Enriched Finite Volume Approximations of the Plane-Parallel Flow at a Small Viscosity

Abstract

We investigate viscous boundary layers of the plane-parallel flow, governed by the stationary Navier–Stokes equations under a certain symmetry. Following the analysis in Gie et al. (Annales de l’Institut Henri Poincaré C. Analyse Non Linéaire, 2018), we first construct the so-called corrector, which is an analytic approximation of the velocity vector field near the boundary. Then, by embedding the corrector function into the classical Finite Volume schemes, we construct the semi-analytic enriched Finite Volume schemes for the plane-parallel flow, and numerically verify that our new enriched schemes reduce significantly the computational error of classical schemes especially near the boundary, and hence produce more accurate approximations without introducing any finer mesh near the boundary.

Introduction

Singular perturbations occur when a small coefficient affects the highest order derivatives in a system of partial differential equations. From the physical point of view, singular perturbations generate thin layers located near the boundary of a domain, called boundary layers where many important physical phenomena occur. In fluid mechanics, the Navier–Stokes equations, which describe the behavior of viscous flows, appear as a singular perturbation of the Euler equations of inviscid flows, where the small parameter is the viscosity or inverse of the Reynolds number. In general, the convergence of the Navier–Stokes solutions to the solution of the Euler (also known as the vanishing viscosity limit problem) remains an outstanding open question in mathematical physics. Because an inviscid flow satisfying the Euler equations is free to slip along the boundary while any viscous flow satisfying the Navier–Stokes must adhere to the boundary, the viscous boundary layers of the Navier–Stokes occur at a small viscosity where a huge gradient of the Navier–Stokes solution is produced. Hence there is a compelling need to develop rigorous analysis as well as efficient computational simulations on the viscous boundary layer problems.

In approximating solutions to singularly perturbed boundary value problems, it is well-known that a very large (discrete) gradient is created near the boundary. Hence, in most classical simulations of singularly perturbed equations, a drastic mesh refinement is usually required near the boundary to obtain an accurate approximation of the solution; a fine mesh of order \(\varepsilon ^{1/2}\), where \(\varepsilon \) is the non-dimensional viscosity parameter, is usually suggested; see, e.g., [22, 23] and other references therein. Departing from massive mesh refinements, new semi-analytic methods have been proposed, see, e.g., [2, 3, 10,11,12,13,14,15,16, 19,20,21,22,23] where the enriched Finite Element (FE) method in a 2D smooth domain or the enriched Finite Volume (FV) method in an interval or a rectangular domain are successfully applied to the convection-reaction-diffusion type scalar equations. The main idea of enriched spaces is adding to the Galerkin basis or the FV space of step functions the specific boundary layer correctors which carry the inherent singularity of the problem. Such methods have proven to be highly efficient without any help of mesh refinement near the boundary where the solution rapidly changes.

In the context of computational fluid mechanics, it is proposed in this article to construct an enriched Finite Volume (FV) scheme to approximate solutions as well as its vorticity to the stationary Navier–Stokes equations (NSE) under a certain symmetry. More precisely, we consider the stationary plane-parallel flow, at a small viscosity, in a periodic channel domain \(\Omega := (0, L)^2 \times (0, 1)\) with boundary at \(z=0,1\). The motion of this 3D stationary plane-parallel flow is governed by the stationary NSE under a symmetry given in the form,

$$\begin{aligned} \varvec{u}^{\varepsilon } = (u^{\varepsilon }_1 (z), \, u^{\varepsilon }_2 (x, z), \, 0 ); \end{aligned}$$
(1.1)

hence the fluid motion occurs in the tangential directions x and y, depending on the tangential variable x and the normal variable z in \(\Omega \). The stationary NSE are then reduced to:

$$\begin{aligned} \left\{ \begin{array}{l} u^{\varepsilon }_1 - \varepsilon {d^2_z u^{\varepsilon }_1}= f_1,\quad \text {in } \Omega ,\\ u^{\varepsilon }_2- \varepsilon {\partial ^2_x u^{\varepsilon }_2}- \varepsilon {\partial ^2_z u^{\varepsilon }_2} + u^{\varepsilon }_1 \, {\partial _x u^{\varepsilon }_2} = f_2,\quad \text {in } \Omega ,\\ u^{\varepsilon }_2 \text { is periodic in}\ x\ \text {with period}\ L,\\ u^{\varepsilon }_i = 0, \quad i=1,2, \text { on } \Gamma , \text { i.e., at } z = 0, 1, \end{array}\right. \end{aligned}$$
(1.2)

provided that the smooth data \(\varvec{f}\), which is periodic in x, satisfies the symmetry,

$$\begin{aligned} \varvec{f} = (f_1 (z), \, f_2 (x, z), \, 0). \end{aligned}$$
(1.3)

Here and throughout this article, we use the notations, \(d_z = d/dz\), \(\partial _x = \partial /\partial _x\), and \(\partial _z = \partial /\partial _z\).

Setting \(\varepsilon = 0\) in the NSE, we find the inviscid limit \(\varvec{u}^0\) of \(\varvec{u}^{\varepsilon }\),

$$\begin{aligned} \varvec{u}^0 = (u^0_1 (z), \, u^0_2 (x, z), \, 0 ), \end{aligned}$$
(1.4)

that satisfies the system:

$$\begin{aligned} \left\{ \begin{array}{l} u^0_1 = f_1, \quad \text {in } \Omega ,\\ u^0_2 + u^0_1 \, {\partial _x u^0_2}= f_2,\quad \text {in } \Omega ,\\ u^0_2 \text { is periodic in}\ x\ \text {with period}\ L. \end{array}\right. \end{aligned}$$
(1.5)

We introduce the vorticity of the Navier–Stokes and Euler solutions,

$$\begin{aligned} \varvec{\omega }^{k}= ( \omega ^k_1(x, z), \omega ^k_2(z), \omega ^k_3(x, z)) := \text {curl }\varvec{u}^{k}= \big (-\partial _z u^k_2, \, d_z u^k_1, \,\partial _x u^k_2\big ),\quad k = \varepsilon , 0. \end{aligned}$$
(1.6)

Now, for the vorticity formulation of the NSE (1.2), we compute \(-\partial _z\)(1.2)\(_2\), \(d_z\)(1.2)\(_1\), and \(\partial _x\)(1.2)\(_2\), and write

$$\begin{aligned} \left\{ \begin{array}{l} \omega ^{\varepsilon }_1 - \varepsilon {\partial ^2_x \omega ^{\varepsilon }_1}- \varepsilon {\partial ^2_z \omega ^{\varepsilon }_1} - \omega ^{\varepsilon }_2 \, \omega ^{\varepsilon }_3 + u^{\varepsilon }_1 \, {\partial _x \omega ^{\varepsilon }_1} = -\partial _z f_2,\quad \text {in } \Omega ,\\ \omega ^{\varepsilon }_2 - \varepsilon {d^2_z \omega ^{\varepsilon }_2}= d_z f_1, \quad \text {in } \Omega ,\\ \omega ^{\varepsilon }_3 - \varepsilon {\partial ^2_x \omega ^{\varepsilon }_3}- \varepsilon {\partial ^2_z \omega ^{\varepsilon }_3} + u^{\varepsilon }_1 \, {\partial _x \omega ^{\varepsilon }_3} = \partial _x f_2, \quad \text {in } \Omega ,\\ \omega ^{\varepsilon }_i \text { is periodic in}\ x\ \text {with period}\ L, i=1,3. \end{array}\right. \end{aligned}$$
(1.7)

To derive a complete set of boundary conditions for the vorticity form of the NSE, we apply the Lighthill principle; see [7, 18]. In fact, by restricting (1.2)\(_{1, 2}\) on \(\Gamma \), and using (1.2)\(_4\), we find the boundary conditions for \(\varvec{\omega }^{\varepsilon }\) so that the two tangential components of vorticity satisfy a non-homogeneous Neumann condition and the normal component satisfies a homogeneous Dirichlet condition:

$$\begin{aligned} \left\{ \begin{array}{l} \partial _z \omega ^{\varepsilon }_1 = \dfrac{1}{\varepsilon } f_2, \quad \text {on } \Gamma ,\\ d_z \omega ^{\varepsilon }_2 = -\dfrac{1}{\varepsilon } f_1, \quad \text {on } \Gamma ,\\ \omega ^{\varepsilon }_3 = 0, \quad \text {on } \Gamma . \end{array}\right. \end{aligned}$$
(1.8)

Repeating the same computations above, we write the vorticity formulation of the Euler equations (EE) (1.5):

$$\begin{aligned} \left\{ \begin{array}{l} \omega ^{0}_1 - \omega ^{0}_2 \, \omega ^{0}_3 + u^{0}_1 \, {\partial _x \omega ^{0}_1} = -\partial _z f_2,\quad \text {in } \Omega ,\\ \omega ^{0}_2 = d_z f_1, \quad \text {in } \Omega ,\\ \omega ^{0}_3 + u^{0}_1 \, {\partial _x \omega ^{0}_3} = \partial _x f_2, \quad \text {in } \Omega ,\\ \omega ^{0}_i \text { is periodic in}\ x\ \text {with period}\ L, i=1,3. \end{array}\right. \end{aligned}$$
(1.9)

Under a sufficient regularity assumption on the data f, e.g., \(f \in H^6(\Omega )\) as stated in Theorem 2.1 below, each system above for \(u^{\varepsilon }\), \(u^0\), \(\omega ^{\varepsilon }\), and \(\omega ^{0}\) possesses a unique solution, which is regular enough for the convergence results stated in Theorem 2.1. In fact, in Sect. 2 below, following the methodology of boundary layer analysis in [5, 7], we first prove the \(L^2\)–convergence of \(\varvec{u}^{\varepsilon }\) to \(\varvec{u}^0\) as well as a certain weak convergence of \(\varvec{\omega }^{\varepsilon }\) to \(\varvec{\omega }^0\) as \(\varepsilon \) tends to 0; here we employ the analysis in [7] where the time-dependent version of (1.2) is considered. In Sect. 3 below, as one of our main tasks in this article, we construct the classical Finite Volume (cFV) approximations as well as our new enriched Finite Volume (eFV) approximations for \(\varvec{u}^{\varepsilon }\), and compare the numerical results of those two schemes, especially when the viscosity \(\varepsilon \) is relatively smaller than the mesh size of the numerical schemes. In the following Sect. 4, we construct and compare the numerical quality of cFV and our new eFV approximations for the vorticity \(\varvec{\omega }^{\varepsilon }\) at a small \(\varepsilon \). Our method of enriched schemes, used in Sects. 3 and 4, is based on the cell-centered Finite Volume settings in, e.g., [1, 4, 6, 8, 9, 16], and it is new in the field of computational fluid mechanics, especially for the non-linear fluid equations (1.2) and (1.7). Note that only simple linear scalar equations have been considered so far in the context of enriched methods and mesh refinement methods; see, e.g., [2, 3, 10,11,12,13,14,15,16, 19,20,21,22,23]. As summarized in Sect. 5, we numerically verify that our novel enriched FV (eFV) scheme reduces significantly the computational error of the classical FV scheme (cFV), especially near the boundary, and hence produces better approximations than the classical one.

Boundary Layer Analysis

We propose an asymptotic expansion,

$$\begin{aligned} \varvec{u}^{\varepsilon } \simeq \varvec{u}^{0} + \varvec{\theta }, \end{aligned}$$
(2.1)

where \(\varvec{\theta }\) is the corrector, which will be explicitly constructed below, in the form,

$$\begin{aligned} \varvec{\theta }= \big (\theta _1 (z), \, \theta _2 (x, z), \, 0 \big ). \end{aligned}$$
(2.2)

Inserting \(\varvec{\theta }\) into the difference between the Eqs. (1.2) and (1.5), we collect all the terms of dominant order \(\varepsilon ^0\). Then we write the asymptotic equation for \(\varvec{\theta }\) as the weakly coupled system below:

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \theta _1 - \varepsilon {d^2_z \theta _1} \simeq 0, \quad \text {in } \Omega ,\\ \displaystyle \theta _2 - \varepsilon \Delta \theta _2 + \big (\theta _1+ u^0_1\big ) {\partial _x \theta _2} \simeq - \theta _1 {\partial _x u^0_2}, \quad \text {in } \Omega ,\\ \theta _i = - u^0_i, \quad \text {on } \Gamma , \quad i=1,2. \end{array}\right. \end{aligned}$$
(2.3)

To construct an (approximate) solution \(\varvec{\theta }\) of (2.3) above, we first introduce a \(C^\infty \) truncation function \(\sigma _L\) (and \(\sigma _R\)) near the boundary at \(z = 0\) (and \(z = 1\)) such that

$$\begin{aligned} \sigma _L(z) = \left\{ \begin{array}{ll} 1, &{} 0 \le z \le 1/4,\\ 0, &{} z \ge 1/2, \end{array}\right. \quad \sigma _R(z) = \sigma _L(1-z). \end{aligned}$$
(2.4)

Then, we define the first component \(\theta _1\) of the corrector as

$$\begin{aligned} \theta _1(z, t) = \sigma _L(z) \, \theta _{1, \, L} (z ) + \sigma _R(z) \, \theta _{1, \, R} (z ), \end{aligned}$$
(2.5)

where

$$\begin{aligned} \left\{ \begin{array}{l} \theta _{1, \, L} (z ) = - u^0_1(0) \, e^{ - z / \sqrt{\varepsilon }}, \\ \theta _{1, \, R} (z )=- u^0_1(1) \, e^{ - (1-z) / \sqrt{\varepsilon }}. \end{array}\right. \end{aligned}$$
(2.6)

One can verify that \({\theta }_1\) enjoys the estimates,

$$\begin{aligned} \Vert {d_z^{m} \theta _1} \Vert _{L^p(\Omega )} \le \kappa \varepsilon ^{\frac{1}{2p} -\frac{m}{2}}, \quad m \ge 0, \quad 1 \le p \le \infty , \end{aligned}$$
(2.7)

as well as the proposed asymptotic equation in (2.3),

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle {\theta _1}- \varepsilon {d^2_z \theta _1}=e.s.t.,\quad \text {in } \Omega ,\\ \displaystyle \theta _1 = - u^0_1, \quad \text {on } \Gamma , \end{array}\right. \end{aligned}$$
(2.8)

where e.s.t. denotes a term that is exponentially small with respect to the small perturbation parameter \(\varepsilon \) in any usual norm in, e.g., \(C^s(\Omega )\) or \(H^s(\Omega )\), \(0 \le s \le \infty \).

Using the first component \(\theta _1\), we define \(\theta _{2, \, L}\) and \(\theta _{2, \, R}\) as the solutions of

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle { \theta _{2, \, L}} - \varepsilon \Delta \theta _{2, \, L} + \big ( \theta _{1, \, L} + u^0_1 \big ) {\partial _x \theta _{2, \, L}} = - \theta _{1, \, L} \, {\partial _x u^0_2} , \quad z > 0,\\ \theta _{2, \, L} \text { is}\ L\text {-periodic in } x,\\ \displaystyle \theta _{2, \, L} = - u^0_2, \quad z = 0, \\ \theta _{2, \, L} \rightarrow 0, \quad \text {as } z \rightarrow \infty , \end{array}\right. \end{aligned}$$
(2.9)

and

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle {\theta _{2, \, R}} -\varepsilon \Delta \theta _{2, \, R} + \big ( \theta _{1, \, R} + u^0_1 \big ) {\partial _x \theta _{2, \, R}} =- \theta _{1, \, R} \, {\partial _x u^0_2} , \quad z < 1,\\ \theta _{2, \, R} \text { is}\ L\text {-periodic in } x,\\ \displaystyle \theta _{2, \, R} = - u^0_2, \quad z = 1, \\ \theta _{2, \, R} \rightarrow 0 \quad \text {as } z \rightarrow -\infty . \end{array}\right. \end{aligned}$$
(2.10)

The elliptic equation (2.9) (or (2.10)) is well-defined as all the coefficients are of class \(C^{\infty }(0, \infty )\) (or \(C^{\infty }(- \infty , 1)\)). Moreover, by performing the energy estimates, one can verify that the \({ \theta _{2, \, L}}\) (or \({ \theta _{2, \, R}}\)) behaves like an exponentially decaying function with respect to the variable \(z/\sqrt{\varepsilon }\) (or \((1-z)/\sqrt{\varepsilon }\)) in the sense that, for \(*=L,R\),

$$\begin{aligned} \left\{ \begin{array}{l} \Vert \partial _x^k \theta _{2, \, *} \Vert _{L^p(\Omega )} \le \kappa \varepsilon ^{\frac{1}{2p}}, \quad k \ge 0, \quad 1 \le p \le \infty , \\ \Vert \partial _x^k \partial _z \theta _{2, \, *} \Vert _{L^1(\Omega )} \le \kappa , \quad k \ge 0, \\ \Vert \partial _x^k \partial _z \theta _{2, \, *} \Vert _{L^2(\Omega )} \le \kappa \varepsilon ^{- \frac{1}{4}}, \quad k \ge 0, \\ \Vert \partial _x^k \partial ^2_z \theta _{2, \, *} \Vert _{L^2(\Omega )} \le \kappa \varepsilon ^{- \frac{3}{4}}, \quad k \ge 0; \end{array}\right. \end{aligned}$$
(2.11)

for the detailed verification of the estimates here, see [7] where the (more involved) time-dependent problem is fully analyzed.

Now, using the truncations in (2.4), we define the second component \(\theta _2\) of the corrector as

$$\begin{aligned} \theta _2(x, z) = \sigma _L(z) \, \theta _{2, \, L} (x, z) + \sigma _R(z) \, \theta _{2, \, R} (x, z), \end{aligned}$$
(2.12)

so that

$$\begin{aligned} \theta _2\ \text {satisfies the estimates in} \ (2.11) \ \text {with}\ \theta _{2, \, *}\ \text {replaced by}\ \theta _2. \end{aligned}$$
(2.13)

In addition, one can verify that the second component \(\theta _2\) satisfies the proposed asymptotic equation in (2.3) up to an exponentially small error:

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle {\theta _2} -\varepsilon \Delta \theta _2 + (\theta _1 + u^0_1) {\partial _x \theta _2} = - \theta _1 {\partial _x u^0_2} + e.s.t., \quad \text {in } \Omega ,\\ \theta _2 = - u^0_2, \quad \text {on } \Gamma . \end{array}\right. \end{aligned}$$
(2.14)

We set

$$\begin{aligned} \overline{\varvec{u}} = \big ( \overline{u}_1 (z), \, \overline{u}_2 (x, z), \, 0 \big ) := \varvec{u}^\varepsilon -(\varvec{u}^0 + \varvec{\theta }), \end{aligned}$$
(2.15)

and let \(\overline{\varvec{\omega }}\) denote the associated corrected vorticity,

$$\begin{aligned} \overline{\varvec{\omega }} = \big ( \overline{\omega }_1 (x,z), \, \overline{\omega }_2 (z), \, \overline{\omega }_3 (x,z) \big ) :=\text {curl } \overline{\varvec{u}} = \big ( - {\partial _z \overline{u}_2}, \, {d_z \overline{u}_1}, \, {\partial _x \overline{u}_2}\big ). \end{aligned}$$
(2.16)

Now, using the equations and estimates on \(\varvec{\theta }\), we state and prove the following convergence results, which agree with those in [7] for the case of time-dependent problem:

Theorem 2.1

Under a sufficient regularity assumption on the data \(\varvec{f}\), e.g., \(\varvec{f} \in H^6(\Omega )\), the difference between the plane-parallel viscous solution and its asymptotic expansion vanishes as the viscosity parameter tends to zero in the sense that

$$\begin{aligned} \left\{ \begin{array}{l} \Vert \overline{{u}}_1 \Vert _{L^2(\Omega )} +\varepsilon ^{\frac{1}{2}} \Vert d_z \overline{{u}}_1 \Vert _{L^2 (\Omega )} \le \kappa \varepsilon ,\\ \Vert \overline{{u}}_2 \Vert _{L^2(\Omega )} + \varepsilon ^{\frac{1}{2}} \Vert \nabla \overline{{u}}_2 \Vert _{L^2 (\Omega )} \le \kappa \varepsilon ,\\ \Vert \overline{\omega }_1 \Vert _{L^2(\Omega )} + \varepsilon ^{\frac{1}{2}} \Vert \nabla \overline{\omega }_1 \Vert _{L^2 (\Omega )} \le \kappa \varepsilon ^{\frac{1}{2}},\\ \Vert \overline{\omega }_2 \Vert _{L^2(\Omega )}+ \varepsilon ^{\frac{1}{2}} \Vert d_z \overline{\omega }_2 \Vert _{L^2 (\Omega )} \le \kappa \varepsilon ^{\frac{3}{4}},\\ \Vert \overline{\omega }_3 \Vert _{L^2(\Omega )} +\varepsilon ^{\frac{1}{2}} \Vert \nabla \overline{\omega }_3 \Vert _{L^2 (\Omega )} \le \kappa \varepsilon . \end{array}\right. \end{aligned}$$
(2.17)

Moreover, as the viscosity \(\varepsilon \) tends to zero, the viscous plane-parallel solution \(\varvec{u}^{\varepsilon }\) converges to the corresponding inviscid solution \(\varvec{u}^0\) in the sense that

$$\begin{aligned} \Vert \varvec{u}^{\varepsilon } - \varvec{u}^0 \Vert _{L^2(\Omega )} \le \kappa \varepsilon ^{\frac{1}{4}}. \end{aligned}$$
(2.18)

We also have

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \big ( \text {curl } \varvec{u}^{\varepsilon }, \, \varvec{\varphi } \big )_{L^2(\Omega )} =\big ( \text {curl } \varvec{u}^{0}, \, \varvec{\varphi } \big )_{L^2(\Omega )} + \big ( \varvec{u}^{0} \times \varvec{n}, \, \varvec{\varphi } \big )_{L^2(\Gamma )}, \quad \forall \varvec{\varphi } \in C(\overline{\Omega }), \end{aligned}$$
(2.19)

which expresses the fact that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \text {curl } \varvec{u}^{\varepsilon } = \text {curl } \varvec{u}^{0} + ( \varvec{u}^{0}|_{\Gamma } \times \varvec{n}) \delta _{\Gamma }, \quad \forall \varvec{\varphi } \in C(\overline{\Omega }), \end{aligned}$$
(2.20)

in the sense of weak\(^*\) convergence of bounded measures on \(\overline{\Omega }\).

Proof

To prove (2.17)\(_{1, 2}\), we write the equation for \(\overline{\varvec{u}}\):

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \overline{u}_1 - \varepsilon {d^2_z \overline{u}_1} = \varepsilon {d^2_z u^0_1} + e.s.t., \displaystyle \text { in } \Omega ,\\ \displaystyle \overline{u}_2 - \varepsilon \Delta \overline{u}_2 + u^\varepsilon _1 {\partial _x \overline{u}_2} + \overline{u}_1 \big ( {\partial _x \theta _2} + {\partial _x u^0_2}\big )=\varepsilon \Delta u^0_2 + e.s.t., \displaystyle \text { in } \Omega ,\\ \displaystyle \overline{\varvec{u}} = 0, \text { on } \Gamma . \end{array}\right. \end{aligned}$$
(2.21)

Multiplying (2.21)\(_1\) by \(\overline{u}_1\) and integrating over \(\Omega \), we find that

$$\begin{aligned} \dfrac{1}{2} \Vert \overline{u}_1 \Vert _{L^2(\Omega )}^2 + \varepsilon \Vert {d_z \overline{u}_1} \Vert _{L^2(\Omega )}^2 \le \kappa \varepsilon ^2 . \end{aligned}$$
(2.22)

For the second component of \(\overline{\varvec{u}}\), we multiply (2.21)\(_2\) by \(\overline{u}_2\) and integrate over \(\Omega \). Then, using (2.13) and (2.22) as well as the fact that \(\Vert \partial _x u^0_2 \Vert _{L^{\infty }(\Omega )}\) is bounded, we find that

$$\begin{aligned}&\Vert \overline{u}_2 \Vert _{L^2(\Omega )}^2 + \varepsilon \Vert \nabla \overline{u}_2 \Vert _{L^2(\Omega )}^2 \nonumber \\&\quad \le \kappa \varepsilon ^{2}+ \dfrac{1}{4} \Vert \overline{u}_2 \Vert _{L^2(\Omega )}^2 + \big \Vert \overline{u}_1 \big ( {\partial _x \theta _2} + {\partial _x u^0_2} \big ) \overline{u}_2 \big \Vert _{L^1(\Omega )}\nonumber \\&\quad \le \kappa \varepsilon ^{2} + \dfrac{1}{4} \Vert \overline{u}_2 \Vert _{L^2(\Omega )}^2 + \Vert \overline{u}_1 \Vert _{L^2(\Omega )} \big \Vert {\partial _x \theta _2} + {\partial _x u^0_2} \big \Vert _{L^{\infty }(\Omega )} \Vert \overline{u}_2 \Vert _{L^2(\Omega )}\nonumber \\&\quad \le \kappa \varepsilon ^{2} + \dfrac{1}{2} \Vert \overline{u}_2 \Vert _{L^2(\Omega )}^2. \end{aligned}$$
(2.23)

Here, after integrating by parts, the integration involved in the third term on the left-hand side of (2.21)\(_2\) vanishes, because \(u^\varepsilon _1\) does not depend on the variable x.

Now, (2.17)\(_{1, 2}\) follows from (2.22) and (2.23).

Thanks to (2.7) and (2.13), (2.18) follows from (2.17)\(_{1, 2}\).

To verify (2.17)\(_{3, 4, 5}\), we first differentiate (2.21)\(_2\) in x and z, and differentiate (2.21)\(_1\) in z. Then, using the expression of \(\varvec{\omega }^\varepsilon \) in (2.16), we find the equations of \(\varvec{\omega }^\varepsilon \):

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle { \overline{\omega }_1} - \varepsilon \Delta \overline{\omega }_1 + u^\varepsilon _1 \, {\partial _x \overline{\omega }_1} \\ \qquad = - \varepsilon {\partial _z (\Delta u^0_2)} +\overline{\omega }_2 \, {\partial _x (u^0_2 + \theta _2)} +\overline{u}_1 \, {\partial _x \partial _z (u^0_2 + \theta _2)} +{d_z u^\varepsilon _1} \, \overline{\omega }_3 +e.s.t., \quad \text {in } \Omega ,\\ \displaystyle { \overline{\omega }_2} - \varepsilon {d^2_z \overline{\omega }_2} = \varepsilon {d^3_z u^0_1} + e.s.t., \quad \text {in } \Omega ,\\ \displaystyle { \overline{\omega }_3} - \varepsilon \Delta \overline{\omega }_3 + u^\varepsilon _1 \, {\partial _x \overline{\omega }_3} =\varepsilon {\partial _x \Delta u^0_2} - \overline{u}_1\, {\partial ^2_x (u^0_2 + \theta _2)} + e.s.t., \quad \text {in } \Omega . \end{array}\right. \end{aligned}$$
(2.24)

Now, restricting (2.21)\(_{1,2}\) on \(\Gamma \), and using (2.21)\(_{3, 4}\), we find the boundary and initial conditions for \(\varvec{\omega }^\varepsilon \):

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle {\partial _z \overline{\omega }_1} = \Delta u^0_2, \quad \text {on } \Gamma , \\ \displaystyle {d_z \overline{\omega }_2} = - {d^2_z u^0_1}, \quad \text {on } \Gamma , \\ \displaystyle \omega ^\varepsilon _3 = 0, \quad \text {on } \Gamma . \end{array}\right. \end{aligned}$$
(2.25)

Here we use the fact that the terms, e.s.t., in (2.21) vanishes on \(\Gamma \); see the right-hand side of (2.8) and (2.14).

Noticing the fact that \(u^\varepsilon _1\) is independent of x, we multiply (2.24)\(_3\) by \(\overline{\omega }_3\), integrate over \(\Omega \), and integrate by parts. Then, using the estimates (2.13) and the regularity of \(\varvec{u}^0\) as well as the convergence result (2.22), we find that

$$\begin{aligned}&\Vert \overline{\omega }_3 \Vert _{L^2(\Omega )}^2 + \varepsilon \Vert \nabla \overline{\omega }_3 \Vert _{L^2(\Omega )}^2 \\&\quad \le \kappa \varepsilon ^{2} + \dfrac{1}{4} \Vert \overline{\omega }_3 \Vert _{L^2(\Omega )}^2 + \big \Vert \overline{u}_1 \big ( {\partial ^2_x \theta _2} + {\partial ^2_x u^0_2} \big ) \overline{\omega }_3 \big \Vert _{L^1(\Omega )}\\&\quad \le \kappa \varepsilon ^{2} + \dfrac{1}{4} \Vert \overline{\omega }_3 \Vert _{L^2(\Omega )}^2 + \Vert \overline{u}_1 \Vert _{L^2(\Omega )} \big \Vert {\partial ^2_x \theta _2} + {\partial ^2_x u^0_2} \big \Vert _{L^{\infty }(\Omega )} \Vert \overline{\omega }_3 \Vert _{L^2(\Omega )}\\&\quad \le \kappa \varepsilon ^{2} + \dfrac{1}{2} \Vert \overline{\omega }_3 \Vert _{L^2(\Omega )}^2. \end{aligned}$$

Hence (2.17)\(_5\) follows.

We multiply (2.24)\(_2\) by \(\overline{\omega }_2\) and integrate over \(\Omega \). Then, using (2.25)\(_2\) and Schwarz, we find that

$$\begin{aligned} \Vert \overline{\omega }_2 \Vert _{L^2(\Omega )}^2 + \varepsilon \Vert {d_z \overline{\omega }_2} \Vert _{L^2(\Omega )}^2\le & {} \kappa \varepsilon ^2 + \dfrac{1}{4} \Vert \overline{\omega }_2 \Vert _{L^2(\Omega )}^2 + \varepsilon \Vert {d^2_z u^0_1} \, \overline{\omega }_2 \Vert _{L^1(\Gamma )}\nonumber \\\le & {} \kappa \varepsilon ^2 + \dfrac{1}{4} \Vert \overline{\omega }_2 \Vert _{L^2(\Omega )}^2 + \varepsilon \Vert {d^2_z u^0_1} \Vert _{L^2(\Gamma )} \Vert \overline{\omega }_2 \Vert _{L^2(\Gamma )}. \end{aligned}$$
(2.26)

Using the regularity of \(\varvec{u}^0\) and the trace theorem, we estimate the 3rd term on the right-hand side of (2.26),

$$\begin{aligned}&\varepsilon \Vert {d^2_z u^0_1} \Vert _{L^2(\Gamma )}\Vert \overline{\omega }_2\Vert _{L^2(\Gamma )}\nonumber \\&\quad \le \kappa \varepsilon \Vert \overline{\omega }_2\Vert _{L^2(\Omega )}^{\frac{1}{2}} \Vert \overline{\omega }_2\Vert _{H^1(\Omega )}^{\frac{1}{2}}\nonumber \\&\quad \le \kappa \varepsilon \Vert \overline{\omega }_2\Vert _{L^2(\Omega )} + \kappa \varepsilon \Vert \overline{\omega }_2\Vert _{L^2(\Omega )}^{\frac{1}{2}} \Vert {d_z \overline{\omega }_2} \Vert _{L^2(\Omega )}^{\frac{1}{2}}\nonumber \\&\quad \le \kappa \varepsilon ^2+\dfrac{1}{8} \Vert \overline{\omega }_2\Vert _{L^2(\Omega )}^2 +\kappa \varepsilon ^{\frac{3}{4}}\Vert \overline{\omega }_2\Vert _{L^2(\Omega )} +\kappa \varepsilon ^{\frac{5}{4}}\Vert {d_z \overline{\omega }_2} \Vert _{L^2(\Omega )}\nonumber \\&\quad \le \kappa \varepsilon ^{\frac{3}{2}} + \dfrac{1}{4} \Vert \overline{\omega }_2\Vert _{L^2(\Omega )}^2 + \dfrac{1}{2} \varepsilon \Vert {d_z \overline{\omega }_2} \Vert _{L^2(\Omega )}^2. \end{aligned}$$
(2.27)

Then (2.17)\(_4\) follows from (2.26) and (2.27):

$$\begin{aligned} \dfrac{1}{2} \Vert \overline{\omega }_2 \Vert _{L^2(\Omega )}^2 + \dfrac{\varepsilon }{2} \Vert {d_z \overline{\omega }_2} \Vert _{L^2(\Omega )}^2 \le \kappa \varepsilon ^{\frac{3}{2}}. \end{aligned}$$
(2.28)

Now, we multiply (2.24)\(_1\) by \(\overline{\omega }_1\) and integrate over \(\Omega \). Then, using (2.25)\(_1\) and Schwarz, we find that

$$\begin{aligned}&\Vert \overline{\omega }_1 \Vert _{L^2(\Omega )}^2 + \varepsilon \Vert \nabla \overline{\omega }_1 \Vert _{L^2(\Omega )}^2\nonumber \\&\quad \le \kappa \varepsilon ^{2} +\dfrac{1}{8} \Vert \overline{\omega }_1 \Vert _{L^2(\Omega )}^2 + \varepsilon \Vert \Delta u^0_2 \Vert _{L^2(\Gamma )} \Vert \overline{\omega }_1\Vert _{L^2(\Gamma )}\nonumber \\&\qquad + \Vert \overline{\omega }_2 \, {\partial _x (u^0_2 + \theta _2)} \, \overline{\omega }_1 \Vert _{L^1(\Omega )} + \bigg | \int _{\Omega } \overline{u}_1 \, {\partial _x \partial _z (u^0_2 + \theta _2)} \, \overline{\omega }_1 \, d\varvec{x}\nonumber \bigg |\\&\qquad + \Vert {d_z u^\varepsilon _1} \, \overline{\omega }_3 \, \overline{\omega }_1 \Vert _{L^1(\Omega )}. \end{aligned}$$
(2.29)

Using the regularity of \(\varvec{u}^0\) and the trace theorem, we estimate the 3rd term on the right-hand side of (2.29),

$$\begin{aligned} \varepsilon \Vert \Delta u^0_2\Vert _{L^2(\Gamma )} \Vert \overline{\omega }_1\Vert _{L^2(\Gamma )}\le & {} \kappa \varepsilon \Vert \overline{\omega }_1\Vert _{L^2(\Omega )}^{\frac{1}{2}} \Vert \overline{\omega }_1\Vert _{H^1(\Omega )}^{\frac{1}{2}}\nonumber \\\le & {} \kappa \varepsilon \Vert \overline{\omega }_1\Vert _{L^2(\Omega )} + \kappa \varepsilon \Vert \overline{\omega }_1\Vert _{L^2(\Omega )}^{\frac{1}{2}} \Vert \nabla \overline{\omega }_1 \Vert _{L^2(\Omega )}^{\frac{1}{2}}\nonumber \\\le & {} \kappa \varepsilon ^2 + \dfrac{1}{16} \Vert \overline{\omega }_1\Vert _{L^2(\Omega )}^2 + \kappa \varepsilon ^{\frac{3}{4}} \Vert \overline{\omega }_1\Vert _{L^2(\Omega )} +\kappa \varepsilon ^{\frac{5}{4}} \Vert \nabla \overline{\omega }_1 \Vert _{L^2(\Omega )}\nonumber \\\le & {} \kappa \varepsilon ^{\frac{3}{2}} +\dfrac{1}{8} \Vert \overline{\omega }_1\Vert _{L^2(\Omega )}^2 +\dfrac{1}{4}\varepsilon \Vert \nabla \overline{\omega }_1 \Vert _{L^2(\Omega )}^2. \end{aligned}$$
(2.30)

Using the regularity of \(\varvec{u}^0\), (2.13), and (2.17)\(_3\), we estimate the 4th term on the right-hand side of (2.29),

$$\begin{aligned} \Vert \overline{\omega }_2 \,{\partial _x (u^0_2 + \theta _2)} \, \overline{\omega }_1\Vert _{L^1(\Omega )}\le & {} \Vert \overline{\omega }_2 \Vert _{L^2(\Omega )} \Vert {\partial _x (u^0_2 + \theta _2)} \Vert _{L^{\infty }(\Omega )} \Vert \overline{\omega }_1 \Vert _{L^2(\Omega )}\nonumber \\\le & {} \kappa \varepsilon ^{\frac{3}{4}} \Vert \overline{\omega }_1 \Vert _{L^2(\Omega )}\nonumber \\\le & {} \kappa \varepsilon ^{\frac{3}{2}} + \dfrac{1}{8} \Vert \overline{\omega }_1\Vert _{L^2(\Omega )}^2. \end{aligned}$$
(2.31)

For the 5th term on the right-hand side of (2.29), since \(\overline{u}_1 = 0\) on \(\Gamma \), by integrating by parts, we write

$$\begin{aligned} \bigg | \int _{\Omega } \overline{u}_1 \, {\partial _x \partial _z (u^0_2 + \theta _2)} \, \overline{\omega }_1 \, d \varvec{x} \bigg | \le \Vert {d_z \overline{u}_1} \, {\partial _x (u^0_2 + \theta _2)} \, \overline{\omega }_1 \Vert _{L^1(\Omega )} + \Vert \overline{u}_1 \, {\partial _x (u^0_2 + \theta _2)} \, {\partial _z \overline{\omega }_1} \Vert _{L^1(\Omega )}. \end{aligned}$$
(2.32)

Then, using the regularity of \(\varvec{u}^0\), (2.13), and (2.17)\(_1\), we find

$$\begin{aligned} \displaystyle \bigg | \int _{\Omega } \overline{u}_1 \, {\partial _x \partial _z (u^0_2 + \theta _2)} \, \overline{\omega }_1 \, d \varvec{x} \bigg |\le & {} \Vert {d_z \overline{u}_1} \Vert _{L^2(\Omega )} \Vert {\partial _x (u^0_2 + \theta _2)} \Vert _{L^{\infty }(\Omega )} \Vert \overline{\omega }_1 \Vert _{L^2(\Omega )}\nonumber \\&+\Vert \overline{u}_1 \Vert _{L^2(\Omega )} \Vert {\partial _x (u^0_2 + \theta _2)} \Vert _{L^{\infty }(\Omega )} \Vert {\partial _z \omega ^\varepsilon _1} \Vert _{L^2(\Omega )}\nonumber \\\le & {} \kappa \varepsilon ^{\frac{1}{2}} \Vert \overline{\omega }_1 \Vert _{L^2(\Omega )} +\kappa \varepsilon \Vert {\partial _z \overline{\omega }_1} \Vert _{L^2(\Omega )} \nonumber \\\le & {} \kappa \varepsilon + \dfrac{1}{8} \Vert \overline{\omega }_1 \Vert _{L^2(\Omega )}^2 + \dfrac{1}{4} \varepsilon \Vert \nabla \omega ^\varepsilon _1 \Vert _{L^2(\Omega )}. \end{aligned}$$
(2.33)

Using (2.17)\(_4\) and (2.7), we estimate the last term on the right-hand side of (2.29),

$$\begin{aligned} \Vert {d_z u^\varepsilon _1} \, \overline{\omega }_3 \, \overline{\omega }_1 \Vert _{L^1(\Omega )}\le & {} \Vert \overline{\omega }_2 \, \overline{\omega }_3 \, \overline{\omega }_1 \Vert _{L^1(\Omega )} + \Vert {d_z u^0_1} \, \overline{\omega }_3 \, \overline{\omega }_1 \Vert _{L^1(\Omega )} + \Vert {d_z \theta _1} \, \overline{\omega }_3 \, \overline{\omega }_1 \Vert _{L^1(\Omega )}\nonumber \\\le & {} \Vert \overline{\omega }_2 \Vert _{L^{\infty }(\Omega )} \Vert \overline{\omega }_3 \Vert _{L^2(\Omega )} \Vert \overline{\omega }_1 \Vert _{L^2(\Omega )}\nonumber \\&+ \big ( \Vert {d_z u^0_1} \Vert _{L^{\infty }(\Omega )} + { \Vert {d_z \theta _1} \Vert _{L^{\infty }(\Omega )} } \big ) \Vert \overline{\omega }_3 \Vert _{L^2(\Omega )} \Vert \overline{\omega }_1 \Vert _{L^2(\Omega )}\nonumber \\\le & {} \big ( \varepsilon \Vert \overline{\omega }_2 \Vert _{L^{\infty }(\Omega )} +\kappa \varepsilon ^{\frac{1}{2}} \big ) \Vert \overline{\omega }_1 \Vert _{L^2(\Omega )}\nonumber \\\le & {} \kappa \varepsilon ^2 \Vert \overline{\omega }_2 \Vert _{L^{\infty }(\Omega )}^2 + \kappa \varepsilon + \dfrac{1}{8} \Vert \overline{\omega }_1 \Vert _{L^2(\Omega )}^2. \end{aligned}$$
(2.34)

We deduce from (2.29)-(2.34) that

$$\begin{aligned} \dfrac{3}{8} \Vert \omega ^\varepsilon _1 \Vert _{L^2(\Omega )}^2 + \dfrac{\varepsilon }{2} \Vert \nabla \omega ^\varepsilon _1 \Vert _{L^2(\Omega )}^2 \le \kappa \varepsilon ^2 \Vert \overline{\omega }_2 \Vert _{L^{\infty }(\Omega )}^2 + \kappa { \varepsilon . } \end{aligned}$$
(2.35)

Since \(\overline{\omega }_2\) is a function in the normal variable z only, using 1D Agmon’s inequality as well as (2.17)\(_3\), we notice that

$$\begin{aligned} \Vert \overline{\omega }_2 \Vert _{L^{\infty }(\Omega )}^2 \le \kappa \Vert \overline{\omega }_2 \Vert _{L^{2}(\Omega )} \Vert { d_z \overline{\omega }_2} \Vert _{L^{2}(\Omega )} \le \varepsilon . \end{aligned}$$
(2.36)

Thus (2.17)\(_3\) follows from (2.35) and (2.36).

Now to verify (2.19), using (2.17), we first notice that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \big ( \text {curl } \varvec{u}^{\varepsilon } - \text {curl } \varvec{u}^{0} - \text {curl } \varvec{\theta } , \, \varphi \big )_{{L}^2(\Omega )} = (0, 0, 0), \end{aligned}$$
(2.37)

for any continuous function \(\varphi \) in \(\overline{\Omega }\). Hence we find that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \big ( \text {curl } (\varvec{u}^\varepsilon - \varvec{u}^0 ), \, \varphi \big )_{{L}^2(\Omega )} = \lim _{\varepsilon \rightarrow 0} \big ( \text {curl } \varvec{\theta }, \varphi \big )_{{L}^2(\Omega )}, \end{aligned}$$

if the limit on the right hand side exists. Hence, to complete the proof, we must show that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \big (\text {curl } \varvec{\theta }, \, \varphi \big )_{{L}^2(\Omega )} = \big ( \varvec{u}^0 \times \varvec{n} , \, \varphi \big )_{{L}^2(\Gamma )}. \end{aligned}$$
(2.38)

After a simple computation, we observe that \(\varvec{u}^0 \times \varvec{n} = ( -u^0_2 , \, u^0_1 , \, 0 )\) at \(z=0\) and \(( u^0_2 , \, - u^0_1 , \, 0 )\) at \(z=1\). Hence what we must show reduces to

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \big (\text {curl } \varvec{\theta }, \, \varphi \big )_{{L}^2(\Omega )} = \big ( u^0_1 - u^0_2, \, \varphi \big )_{L^2(\Gamma _0)} + \big ( - u^0_1 + u^0_2, \, \varphi \big )_{L^2(\Gamma _1)}, \end{aligned}$$
(2.39)

where \(\Gamma _k = \Gamma |_{z=k}\) with \(k= 0, 1\).

Thanks to (2.13), we notice that the limit of \(\text {curl } \varvec{\theta }\) at \(\varepsilon =0\) (the left hand side of (2.39)) must exist in the space of Radon measures. Now, to find the explicit form of this limit, we recall that

$$\begin{aligned} \text {curl } \varvec{\theta } = \big ( - {\partial _z \theta _2}, \, {d_z \theta _1}, \, {\partial _x \theta _2} \big ). \end{aligned}$$
(2.40)

Using (2.13), we notice that the third component of (2.40) satisfies that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \big ( {\partial _x \theta _2}, \, \varphi \big ) = 0. \end{aligned}$$
(2.41)

For the second component of (2.40), we use (2.5) and write

$$\begin{aligned} \big ( {d_z \theta _1} , \, \varphi \big )_{L^2(\Omega )} =\big ( {d_z \theta _{1, \, L}} \sigma _L , \, \varphi \big )_{L^2(\Omega )} + \big ( {d_z \theta _{1, \, R}} \sigma _R , \, \varphi \big )_{L^2(\Omega )} + e.s.t. \end{aligned}$$

We notice from (2.6) that

$$\begin{aligned} d_z \theta _{1, \, L} = \dfrac{1}{\sqrt{\varepsilon }} \, u^0_1(0) \, e^{-z/\sqrt{\varepsilon }}, \quad \varepsilon >0, \end{aligned}$$
(2.42)

forms approximations to the identity (multiplied by \(u^0_1(0)\)) at \(z = 0\). Hence, we find that

$$\begin{aligned} \displaystyle \lim _{\varepsilon \rightarrow 0} \big ( {d_z \theta _{1, \, L}} , \, \sigma _L \, \varphi \big )_{L^2(\Omega )} = \int _{[0, \, L]^2} u^0_1(0) \, \varphi (x, y, 0) \, dxdy =\big ( u^0_1, \, \varphi \big )_{L^2(\Gamma _0)}. \end{aligned}$$

Similarly, one can also verify that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \big ( {d_z \theta _{1, \, R}} , \, \sigma _R \, \varphi \big )_{L^2(\Omega )} = - \big ( u^0_1, \, \varphi \big )_{L^2(\Gamma _1)}. \end{aligned}$$

Hence we obtain that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \big ( {d_z \theta _{1}} , \, \varphi \big )_{L^2(\Omega )} = \big ( u^0_1, \, \varphi \big )_{L^2(\Gamma _0)} - \big ( u^0_1, \, \varphi \big )_{L^2(\Gamma _1)}, \quad \forall \varphi \in C(\overline{\Omega }). \end{aligned}$$
(2.43)

For the first component of (2.40), we set \(\theta _2 = {\Phi } + ({\theta }_2 - {\Phi })\) where

$$\begin{aligned} \left\{ \begin{array}{l} \Phi (x, z) = \sigma _L(z) \, \Phi _{ L} (x, z) + \sigma _R(z) \, \Phi _{ R} (x, z),\\ \Phi _{ L} (z, t) = - u^0_2(x, 0) \, e^{ - z / \sqrt{\varepsilon }}, \\ \Phi _{R} (z, t) = - u^0_2(x, 1) \, e^{ - (1-z) / \sqrt{\varepsilon }}; \end{array}\right. \end{aligned}$$
(2.44)

hence \({\theta }_2 = {\Phi }\) on \(\Gamma \). Under this setting, we write

$$\begin{aligned} \big ( - {\partial _z \theta _{2}} , \, \varphi \big )_{L^2(\Omega )} =\big ( - {\partial _z \Phi } , \, \varphi \big )_{L^2(\Omega )} +\big ( - {\partial _z({\theta }_2 - {\Phi })} , \, \varphi \big )_{L^2(\Omega )}. \end{aligned}$$

By the exactly same argument as for \(\theta _1\), one can verify that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} ( - {\partial _z \Phi } , \, \varphi )_{L^2(\Omega )} =-\big ( u^0_2, \, \varphi \big )_{L^2(\Gamma _0)} +\big ( u^0_2, \, \varphi \big )_{L^2(\Gamma _1)}, \quad \forall \varphi \in C(\overline{\Omega }). \end{aligned}$$
(2.45)

On the other hand, because the \(L^1\) norm of \(\partial ({\theta }_2 -{\Phi }) / \partial z\) is bounded independently in \(\varepsilon \) by (2.7) with \(\theta _1\) replaced by \(\Phi \) and (2.11)\(_2\), there exist a subsequence \(\varepsilon ^{\prime }\) of \(\varepsilon \), and \(\mu \) in the dual of \(C(\overline{\Omega })\) such that

$$\begin{aligned} \lim _{\varepsilon ^{\prime } \rightarrow 0} \big ( - {\partial _z ({\theta }_2 - {\Phi })} , \, \varphi \big )_{L^2(\Omega )} = \big ( \mu , \, \varphi \big )_{L^2(\Omega )}. \end{aligned}$$
(2.46)

Especially, taking the test function \(\varphi \) in \(C(\overline{\Omega }) \cap H^1\), thanks to (2.45), and the earlier weak convergence result of the vorticity in [17], we find that \(\mu \) in (2.46) must to be 0. Here we use the fact that \((C(\overline{\Omega }))^{\prime } \subset \big (C(\overline{\Omega }) \cap H^1(\Omega )\big )^{\prime }\). Moreover, since the limit is unique, it has to be attained for the whole sequence:

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \big ( - {\partial _z ({\theta }_2 - {\Phi })} , \, \varphi \big )_{L^2(\Omega )} = 0, \quad \forall \varphi \in C(\overline{\Omega }). \end{aligned}$$

In conclusion, we observe that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \big ( - {\partial _z \theta _{2}} , \, \varphi \big )_{L^2(\Omega )} = \big ( u^0_2, \, \varphi \big )_{L^2(\Gamma _0)} - \big ( u^0_2, \, \varphi \big )_{L^2(\Gamma _1)}, \quad \forall \varphi \in C(\overline{\Omega }). \end{aligned}$$
(2.47)

Finally, (2.19) follows from (2.39), (2.41), (2.43), and (2.47), and the proof of Theorem 2.1 is now complete. \(\square \)

FV Approximations of the Velocity Vector Field

We construct the Finite Volume (FV) approximations of the solution \(\varvec{u}^{\varepsilon }\) of (1.2) (defined in (1.1)) when the viscosity parameter \(\varepsilon \) is small. The data \(\varvec{f}\) in (1.3) is assumed (and will be chosen) to be sufficiently regular for the computations below.

In Sect. 3.1, we first recall the classical FV scheme for the velocity and perform the numerical computations to observe that the dominant error in approximating \(\varvec{u}^{\varepsilon }\) occurs near the boundary at \(z = 0, 1\), due to the effect boundary layers when the viscosity \(\varepsilon \) is relatively small with respect to the mesh size h, i.e., \(\varepsilon ^{1/2} < h\). Next, in Sect. 3.2, we enrich the classical FV scheme by adding (an explicit modification of) the corrector \(\varvec{\theta }\), defined in (2.2), as an additional basis function into the classical FV (cFV) space of step functions. Performing numerical computations, we verify below that our novel enriched FV (eFV) scheme reduces significantly the computational error near the boundary \(\Gamma \) as the boundary layer corrector well approximates \(\varvec{u}^{\varepsilon }\) near \(\Gamma \). The computational error of the classical FV (cFV) and the enriched FV (eFV) schemes are compared in Sect. 3.3, from which we confirm that our eFV works better than the classical one.

Classical FV (cFV) Approximations of the Velocity Vector Field

We briefly construct in this section the classical Finite Volume (cFV) approximations of \(\varvec{u}^{\varepsilon }\) of (1.2) for a fixed (small) viscosity \(\varepsilon \).

Noticing that the Eq. (1.2) and the data are independent from the y variable, and periodic in the x variable, we introduce the computational domain in the z variable,

$$\begin{aligned} \Omega _z = \{ 0< z < 1 \} \subset \mathbb {R}_{z}, \end{aligned}$$
(3.1)

in which we aim to approximate the velocity \(\varvec{u}^{\varepsilon } =(u^{\varepsilon }_1(z), \, u^{\varepsilon }_2(x, z) )\). To manage the dependency of \(u^{\varepsilon }_2\) on the periodic variable x, we will use the partial Fourier transform in x as described in details below.

Now we recall the 1D cell-centered cFV setting in \(\Omega _z\) from, e.g., [8, 9]:

We first fix a partition size \(N >0\) of \(\Omega _z= (0, \,1)\). Choosing \(N + 1\) nodal points \(z_{k+1/2}\), \(0 \le k \le N\), we write a partition of \(\Omega _z\) in the form,

$$\begin{aligned} 0 = z_{\frac{1}{2}}< z_{\frac{3}{2}}< \cdots< z_{N-\frac{1}{2}} < z_{N + \frac{1}{2}} =1. \end{aligned}$$
(3.2)

We also choose two fictitious points outside of \(\Omega _z \),

$$\begin{aligned} z_{-\frac{1}{2}} = - z_{ \frac{3}{2}}, \qquad z_{N + \frac{3}{2}} = z_{N + \frac{1}{2}} + (z_{N + \frac{1}{2}} - z_{N - \frac{1}{2}}) = 2 - z_{N - \frac{1}{2}}. \end{aligned}$$
(3.3)

We define the 1D control volumes \(I_{k}\) of \(\Omega _z\) as

$$\begin{aligned} I_{k} = (z_{k - \frac{1}{2}}, \, z_{k + \frac{1}{2}}), \quad 1 \le k \le N, \quad \text {i.e., } \cup _{k=1}^N I_k = \Omega _z, \end{aligned}$$
(3.4)

as well as two fictitious cells (or intervals) near, but outside \(\Omega _z\), as

$$\begin{aligned} I_{0} = (z_{- \frac{1}{2}}, \, z_{ \frac{1}{2}}), \qquad \qquad I_{N+1} = (z_{N + \frac{1}{2}}, \, z_{N + \frac{3}{2}}). \end{aligned}$$
(3.5)

Note that \(I_0\) and \(I_{N+1}\) are introduced to enforce a proper boundary condition for the FV functions; see (3.9), (3.21), (3.36), (4.2), (4.8), (4.11), and (4.15) below.

The cell center \(z_k\) of \(I_k\) is then defined by

$$\begin{aligned} z_k = \dfrac{1}{2} (z_{k - \frac{1}{2}} + z_{k + \frac{1}{2}}), \quad 0 \le k \le N+1, \end{aligned}$$
(3.6)

and we introduce the intervals between the cell centers,

$$\begin{aligned} I_{k+\frac{1}{2}} = ( z_{k}, \, z_{k+1} ), \quad 0 \le k \le N. \end{aligned}$$
(3.7)

We also set the size of each interval as

$$\begin{aligned} h_k =z_{k + \frac{1}{2}} - z_{k - \frac{1}{2}}, \, 0 \le k \le N+1, \qquad h_{k+ \frac{1}{2}} = z_{k + 1} - z_{k}, \, 0 \le k \le N. \end{aligned}$$
(3.8)

Using the discretization in the z direction above, we define the 1D FV space of step functions in \(\Omega _z = (0, 1)\), supplemented with the zero Dirichlet boundary condition, in the form,

$$\begin{aligned} V_h = \Big \{ u_h = \sum _{k= 0}^{N+1} u_k \chi _{I_k}, \, \, u_k \in \mathbb {R} \Big | \, { u_0 = - u_1, \, \, \, u_{N+1} = - u_N } \Big \}, \end{aligned}$$
(3.9)

where \(\chi _E\) is the characteristic function on a set E.

We define the discrete FV derivative in the z direction for a step function \(u_h\) in \(V_h\) by

$$\begin{aligned} \nabla _h u_h = \sum _{k=0}^{N} \bigg ( \dfrac{u_{k+1} - u_k}{h_{k+ \frac{1}{2}}} \bigg ) \chi _{I_{k+\frac{1}{2}}}. \end{aligned}$$
(3.10)

The 1D FV space \(V_h\) is equipped with the following scalar products \((\cdot , \, \cdot )_{V_h}\) and \(((\cdot , \, \cdot ))_{V_h}\) which mimic those of the \(L^2(\Omega _z)\) and \(H^1_0(\Omega _z)\):

For \(u_h\), \(v_h \in V_h\),

$$\begin{aligned} (u_h, \, v_h)_{V_h} = \sum _{k=1}^{N} u_k v_k h_k, \qquad ((u_h, \, v_h))_{V_h} = \sum _{k=1}^{N} \dfrac{ (u_{k+1} - u_k) (v_{k+1} - v_k)}{h_{k+ \frac{1}{2}}} . \end{aligned}$$
(3.11)

The corresponding norms \(| \cdot |_{V_h}\) and \(\Vert \cdot \Vert _{V_h}\) are defined as well.

Classical FV (cFV) approximation of the velocity \(\varvec{u}^{\varepsilon }\)

Now we build a cFV scheme to approximate a solution \(\varvec{u}^{\varepsilon }\) to (1.2):

For the first component \(u^{\varepsilon }_1 = u^{\varepsilon }_1(z)\) of \(\varvec{u}^{\varepsilon }\), we integrate the equation (1.2)\(_1\) over each \(I_k\), \(1 \le k \le N\), and integrate by parts the viscous term to write,

$$\begin{aligned} \int _{I_k} u^{\varepsilon }_1 \, dz -\varepsilon \Big ( d_z u^{\varepsilon }_1 (z_{k +\frac{1}{2}}) - d_z u^{\varepsilon }_1 (z_{k - \frac{1}{2}}) \Big ) = \int _{I_k} f_1 \, dz. \end{aligned}$$
(3.12)

Projecting the equation into the FV space \(V_h\) and using (3.10), we obtain the following discrete equation:

Find\(u_{1, h} = \sum _{k=0}^{N+1} u_{k} \chi _{I_k} \in V_h\)such that

$$\begin{aligned} -\dfrac{\varepsilon }{h_{k- \frac{1}{2}}} u_{k-1} + \bigg ( h_k + \dfrac{\varepsilon }{h_{k- \frac{1}{2}}} + \dfrac{\varepsilon }{h_{k+ \frac{1}{2}}} \bigg )u_{k} -\dfrac{\varepsilon }{h_{k+ \frac{1}{2}}} u_{k+1} = f_1(z_k) h_k, \quad 1 \le k \le N. \end{aligned}$$
(3.13)

For the second component \(u^{\varepsilon }_2 = u^{\varepsilon }_2(x, z)\) of \(\varvec{u}^{\varepsilon }\), thanks to the periodicity in x with period L, we introduce the partial Fourier expansion in x, up to the mode \(N_x>0\),

$$\begin{aligned} u^{\varepsilon }_2 (x, z) \cong \sum _{|j| \le N_x} {u}_{2}^j(z) e^{2 \pi j i x / L}. \end{aligned}$$
(3.14)

Then, by taking the Fourier transform in x of (1.2), we write the equation of \(u^{\varepsilon }_{2}\) mode by mode:

$$\begin{aligned} \left\{ \begin{array}{l} \Big (1 + \varepsilon (2 \pi j/L)^2 + i (2 \pi j/L) u^{\varepsilon }_1(z) \Big ) {u}_{2}^{j} - \varepsilon d^2_z {u}_{2}^{j} = {f}_{2}^{j}, \quad 0< z < 1,\\ \qquad {u}_{2}^{ j} = 0, \quad z = 0, 1, \end{array}\right. \end{aligned}$$
(3.15)

for each \(-N_x \le j \le N_x\). Here \({f}_{2}^{j} = {f}_{2}^{j}(z)\) is the j-th Fourier coefficient of \(f_2\).

Applying the 1D cFV discretization as in (3.12)–(3.16), we write the discrete equation for each \({u}_{2}^{j}\):

Find\(u_{2, h}^j = \sum _{k=0}^{N+1} u_{k}^j \chi _{I_k} \in V_h\), \(-N_x \le j \le N_x\), such that

$$\begin{aligned} -\dfrac{\varepsilon }{h_{k- \frac{1}{2}}} u_{k-1}^j + \bigg ( A^j_k \, h_k + \dfrac{\varepsilon }{h_{k- \frac{1}{2}}} + \dfrac{\varepsilon }{h_{k+ \frac{1}{2}}} \bigg )u_{k}^j -\dfrac{\varepsilon }{h_{k+ \frac{1}{2}}} u_{k+1}^j = f_2^j(z_k) h_k, \quad 1 \le k \le N. \end{aligned}$$
(3.16)

Here we set

$$\begin{aligned} A^j_k = 1 + \varepsilon (2 \pi j/L)^2 + i (2 \pi j/L) u_{k}, \end{aligned}$$
(3.17)

by using the cFV solution \(u_{1, h} = \sum _{k=0}^{N+1} u_{k} \chi _{I_k}\) of (3.13).

Once we compute the cFV solutions \(u_{2, h}^j\) for \(-N_x \le j \le N_x\), then the cFV solution \(u_{2, h}\) of \(u_{2}^{\varepsilon }\) is obtained by

$$\begin{aligned} u_{2, h} = \sum _{|j| \le N_x} {u}_{2, h}^j e^{2 \pi j i x / L}. \end{aligned}$$
(3.18)

In conclusion, by solving sequentially (3.13) and (3.16), we obtain the classical FV (cFV) solution \((u_{1, h}, \, u_{2, h})\), which approximates the solution \(\varvec{u}^{\varepsilon }\) of (1.2) at a fixed viscosity \(\varepsilon \).

For numerical computations below in Sects. 3.3 and 4.3, the dominant computational error occurs always near the boundaries at \(z=0, 1\), and hence we choose a reasonably large number \(N_x\) of Fourier modes as \(N_x = \min (N/2, 60)\).

Enriched FV (eFV) Approximations of the Velocity Vector Field

Here we enrich the cFV space \(V_h\) by adding (proper approximation of) the boundary layer corrector, \(\varvec{\theta } = \big (\theta _1 (z), \, \theta _2 (x, z), \, 0 \big )\), defined in (2.2) and (2.3).

Enriched FV (eFV) approximation of \({u}^{\varepsilon }_1\)

In order to approximate \(u^{\varepsilon }_1\) especially when the viscosity \(\varepsilon \) is small, we normalize the boundary values of \( \theta _{1, \, L} (z ) \) and \( \theta _{1, \, R} (z ), \) defined in (2.6), and introduce the exponentially decaying functions,

$$\begin{aligned} \overline{\theta }_{1, \, L} (z ) = e^{ - z / \sqrt{\varepsilon }}, \qquad \overline{\theta }_{1, \, R} (z ) = e^{ - (1-z) / \sqrt{\varepsilon }}, \end{aligned}$$
(3.19)

that satisfy

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \overline{\theta }_{1, \, *} - \varepsilon {d^2_z \overline{\theta }_{1, \, *}} = 0, \quad \text {in } \Omega _z = (0, \, 1), \quad * = L, R, \\ \displaystyle \overline{\theta }_{1, \, L} = 1, \quad \text {at } z = 0, \\ \displaystyle \overline{\theta }_{1, \, R} = 1, \quad \text {at } z = 1. \end{array}\right. \end{aligned}$$
(3.20)

Here we used, for computational convenience, the exponentially decaying functions \(\theta _{1, \, L}\) and \(\theta _{1, \, R}\) instead of the real corrector \(\theta _{1}\) as in (2.5) because the effect of \(\theta _{1, \, L}\) (or \(\theta _{1, \, R}\)) on the boundary at \(z = 1\) (or \(z = 0\)) is exponentially small with respect to the small viscosity, and hence it is negligible compared to the computational error.

We define the enriched FV (eFV) space \(\widetilde{V}_{1, h}\) for \(u^{\varepsilon }_1\) by adding into the classical FV space \(V_h\) the basis functions \( \overline{\theta }_{1, \, L} \) and \( \overline{\theta }_{1, \, R} \) :

$$\begin{aligned} \widetilde{V}_{1, h} = \left\{ \begin{array}{l} \displaystyle \widetilde{u}_{1,h} =\sum _{k= 0}^{N+1} \widetilde{u}_{k} \chi _{I_k} + r_L \overline{\theta }_{1, \, L} (z) + r_R \overline{\theta }_{1, \, R} (z) , \, \, \widetilde{u}_{k}, \, r_L, \, r_R \in \mathbb {R}, \\ \text {such that } \,\, { \widetilde{u}_{0} = -2 r_{L} -\widetilde{u}_{1}, \, \, \, \widetilde{u}_{N+1} = - \widetilde{u}_{N} -2 r_{R} . } \end{array}\right\} . \end{aligned}$$
(3.21)

Note that the constraints in the definition of \(\widetilde{V}_{1, h}\) enforce the zero Dirichlet boundary condition from the Eq. (1.2) for \(u^{\varepsilon }_1\). That is, e.g., at \(z = 0\),

$$\begin{aligned} \widetilde{u}_{1,h}|_{z= 0} = \dfrac{\widetilde{u}_{ 0} + \widetilde{u}_{ 1}}{2} + r_L = 0. \end{aligned}$$
(3.22)

Modifying the FV derivative in (3.10), we define the enriched FV derivative in the z direction for a function \(\widetilde{u}_{1, h}\) in \(\widetilde{V}_{1, h}\) by

$$\begin{aligned} \widetilde{\nabla }_h \widetilde{u}_{1,h} = \sum _{k=0}^{N} \bigg ( \dfrac{\widetilde{u}_{k+1} - \widetilde{u}_{k}}{h_{k+ \frac{1}{2}}} \bigg ) \chi _{I_{k+\frac{1}{2}}} + r_L d_z \overline{\theta }_{1, \, L} (z) + r_R d_z \overline{\theta }_{1, \, R} (z) , \end{aligned}$$
(3.23)

Now, to construct the enriched FV (eFV) approximation of \(u^{\varepsilon }_1\), we consider the \(L^2\) inner products of the Eq. (1.2)\(_1\) against the basis functions \(\overline{\theta }_{1, \, L }\), \(\overline{\theta }_{1, \, R}\), and \(\chi _{I_{k}}\), \(1 \le k \le N\):

(i):

We multiply the Eq. (1.2)\(_1\) by \(\overline{\theta }_{1, \, L }\) and integrate over \(I_{1} = (0, \, z_{3/2})\) as the effect of \(\overline{\theta }_{1, \, L }\) is negligible outside of \(I_{1}\). Integrating the diffusive term by parts, we obtain,

$$\begin{aligned}&\int _{I_1} u^{\varepsilon }_1 \overline{\theta }_{1, \, L } \, dz -\varepsilon \Big ( (d_z u^{\varepsilon }_1 \overline{\theta }_{1, \, L }) (z_{ \frac{3}{2}}) -(d_z u^{\varepsilon }_1 \overline{\theta }_{1, \, L }) (0) \Big )\nonumber \\&+ \varepsilon \int _{I_1} d_z u^{\varepsilon }_1 d_z \overline{\theta }_{1, \, L } \, dz =\int _{I_1} f_1 \overline{\theta }_{1, \, L } \, dz. \end{aligned}$$
(3.24)
(ii):

Multiplying (1.2)\(_2\) by \(\overline{\theta }_{1, \, R}\), integrating over \(I_{N} = (z_{N - 1/2}, \, 1)\), and integrating the diffusive term by parts, we similarly obtain,

$$\begin{aligned}&\int _{I_N} u^{\varepsilon }_1 \overline{\theta }_{1, \, R } \, dz -\varepsilon \Big ( (d_z u^{\varepsilon }_1 \overline{\theta }_{1, \, R }) (1) - (d_z u^{\varepsilon }_1 \overline{\theta }_{1, \, R }) (z_{N-\frac{1}{2}}) \Big )\nonumber \\&+ \varepsilon \int _{I_N} d_z u^{\varepsilon }_1 d_z \overline{\theta }_{1, \, R } \, dz =\int _{I_N} f_1 \overline{\theta }_{1, \, R } \, dz. \end{aligned}$$
(3.25)
(iii):

Multiplying (1.2)\(_2\) by \(\chi _{I_k}\), \(1 \le k \le N\), integrating over \(\Omega _z = (0, \, 1)\), and integrating the diffusive term by parts, we find that

$$\begin{aligned} \int _{I_k} u^{\varepsilon }_1 \, dz -\varepsilon \Big ( d_z u^{\varepsilon }_1 (z_{k +\frac{1}{2}}) - d_z u^{\varepsilon }_1 (z_{k - \frac{1}{2}}) \Big ) =\int _{I_k} f_1 \, dz, \qquad 1 \le k \le N. \end{aligned}$$
(3.26)

Because the exponentially decaying functions \(\overline{\theta }_{1, \, * }\), \(* = L, R\), satisfy the Eq. (3.20)\(_1\), the projections of the Eqs. (3.24)–(3.26) into the enriched space \(\widetilde{V}_{1, h}\) are the same as those with \(u^{\varepsilon }_1\) replaced by \(\sum _{k= 0}^{N+1} \widetilde{u}_{k} \chi _{I_k}\). Using (3.23) and the values \(\widetilde{u}_{1,0}\) and \(\widetilde{u}_{1, N+1}\) in the setting of \(\widetilde{V}_{1,h}\) as well, we obtain the following projections of (3.24)–(3.26) onto the enriched space \(\widetilde{V}_{1, h}\):

Find\(\widetilde{u}_{1, h} \in \widetilde{V}_{1,h}\)such that

$$\begin{aligned}&\displaystyle -\dfrac{\varepsilon }{h_{\frac{1}{2}}} \Big ( 1 + \dfrac{1}{2} \int _{I_1} d_z \overline{\theta }_{1, \, L } \, dz \Big ) \widetilde{u}_{0} \nonumber \\&\quad + \bigg [ \int _{I_1} \overline{\theta }_{1, \, L } \, dz +\dfrac{\varepsilon }{h_{\frac{3}{2}}} \overline{\theta }_{1, \, L }(z_{\frac{3}{2}}) +\dfrac{\varepsilon }{h_{\frac{1}{2}}} -\dfrac{\varepsilon }{2} \Big ( \dfrac{1}{h_{\frac{3}{2}}} - \dfrac{1}{h_{\frac{1}{2}}} \Big ) \int _{I_1} d_z \overline{\theta }_{1, \, L } \, dz \bigg ] \widetilde{u}_{1}\nonumber \\&\quad -\dfrac{\varepsilon }{h_{\frac{3}{2}}} \Big ( \overline{\theta }_{1, \, L }(z_{\frac{3}{2}}) - \dfrac{1}{2} \int _{I_1} d_z \overline{\theta }_{1, \, L } \, dz \Big ) \widetilde{u}_{2} = f_1(z_1) \int _{I_1} \overline{\theta }_{1, \, L } \, dz, \end{aligned}$$
(3.27)
$$\begin{aligned}&\quad -\dfrac{\varepsilon }{h_{k- \frac{1}{2}}} \widetilde{u}_{k-1} + \bigg ( h_k + \dfrac{\varepsilon }{h_{k- \frac{1}{2}}} + \dfrac{\varepsilon }{h_{k+ \frac{1}{2}}} \bigg ) \widetilde{u}_{k} -\dfrac{\varepsilon }{h_{k+ \frac{1}{2}}} \widetilde{u}_{k+1} = f_1(z_k) h_k, \quad 1 \le k \le N,\nonumber \\ \end{aligned}$$
(3.28)

and

$$\begin{aligned}&\displaystyle -\dfrac{\varepsilon }{h_{N-\frac{1}{2}}} \Big ( \overline{\theta }_{1, \, R }(z_{N-\frac{1}{2}}) + \dfrac{1}{2} \int _{I_N} d_z \overline{\theta }_{1, \, R } \, dz \Big ) \widetilde{u}_{N-1}\nonumber \\&\quad + \bigg [ \int _{I_N} \overline{\theta }_{1, \, R } \, dz +\dfrac{\varepsilon }{h_{N+\frac{1}{2}}} +\dfrac{\varepsilon }{h_{N-\frac{1}{2}}} \overline{\theta }_{1, \, R }(z_{N-\frac{1}{2}}) -\dfrac{\varepsilon }{2} \Big ( \dfrac{1}{h_{N+\frac{1}{2}}} - \dfrac{1}{h_{N-\frac{1}{2}}} \Big ) \int _{I_N} d_z \overline{\theta }_{1, \, R } \, dz \bigg ] \widetilde{u}_{N}\nonumber \\&\quad -\dfrac{\varepsilon }{h_{N+\frac{1}{2}}} \Big ( 1 - \dfrac{1}{2} \int _{I_N} d_z \overline{\theta }_{1, \, R} \, dz \Big ) \widetilde{u}_{N+1} = f_1(z_N) \int _{I_N} \overline{\theta }_{1, \, R } \, dz. \end{aligned}$$
(3.29)

Solving the equations above, we obtain the eFV approximation \(\widetilde{u}_{1, h}\) as a solution of (3.27)–(3.28), because the unknowns \(r_L\) and \(r_R\) are also determined by (3.22) in the setting of \(\widetilde{V}_{1,h}\).

Enriched FV (eFV) approximation of \({u}^{\varepsilon }_2\)

For the second component \(u^{\varepsilon }_2 = u^{\varepsilon }_2(x, z)\) of \(\varvec{u}^{\varepsilon }\), we recalled from (3.14) the the partial Fourier expansion in x, up to the mode \(N_x>0\),

$$\begin{aligned} u^{\varepsilon }_2 (x, z) \cong \sum _{|j| \le N_x} {u}_{2}^j(z) e^{2 \pi j i x/ L}. \end{aligned}$$
(3.30)

and construct first below the enriched FV (eFV) approximation for each \({u}_{2}^j\), which satisfies (3.15).

To find a proper corrector function for \({u}_{2}^j\) (near \(z=0\) or \(z=1\)), taking the partial Fourier transform in x of (2.9)–(2.10), we write the asymptotic equations of \(\theta ^j_{2, *}\), \(*=L, R\), at each Fourier mode j,

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \widetilde{A}^j_1 \theta ^j_{2, L} -\varepsilon d_z^2 (\theta ^j_{2, L}) = - i (2 \pi j / L) r_{L} e^{-z/\sqrt{\varepsilon }} (u^0_2)^j , \quad z> 0, \\ \widetilde{A}^j_N \theta ^j_{2, R} -\varepsilon d_z^2 (\theta ^j_{2, R}) = - i (2 \pi j /L) r_{R} e^{-(1-z)/\sqrt{\varepsilon }} (u^0_2)^j , \quad z< 1. \end{array}\right. \end{aligned}$$
(3.31)

Here \(\widetilde{A}^j_1\) and \(\widetilde{A}^j_N\) are defined by

$$\begin{aligned} \widetilde{A}^j_k = 1 + \varepsilon (2 \pi j / L)^2 + i (2 \pi j / L) \widetilde{u}_{ k}, \quad 1 \le k \le N, \end{aligned}$$
(3.32)

provided that the eFV solution \(\widetilde{u}_{1, h}\) for the first component \(u^{\varepsilon }_1\) is already computed in advance. The coefficients \(r_L\) and \(r_R\) are also computed by the eFV of \(u^{\varepsilon }_1\), and \((u^0_2)^j\) denotes the j-th Fourier coefficient of the limit solution \(u^0_2\). On the right-hand side of (3.31), we dropped the lower order term \(-\varepsilon (2 \pi j / L)^2 (u^0_2)^j\), because the contribution of this lower order term is small and negligible compared to the computational error.

We supplement the Eq. (3.31) with the normalized boundary conditions,

$$\begin{aligned} \theta ^j_{2, L}(0) = \theta ^j_{2, R}(1) = 1. \end{aligned}$$
(3.33)

Now, aiming to construct a simple eFV scheme, we temporarily ignore the effect of the right-hand side of (3.31), and introduce the homogeneous solutions \(\Theta _{L}^j\) and \(\Theta _{R}^j\) of (3.31)–(3.33) that satisfy

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \widetilde{A}^j_1 \Theta _{L}^j -\varepsilon d_z^2 \Theta _{L}^j = 0 , \quad z> 0, \\ \widetilde{A}^j_N \Theta ^j_{ R} -\varepsilon d_z^2 \Theta ^j_{ R} = 0 , \quad z< 1, \\ \Theta ^j_{L}(0) = \Theta ^j_{R}(1) = 1. \end{array}\right. \end{aligned}$$
(3.34)

The explicit expressions of \(\Theta _{*}^j\), \(*=L, R\), are given by

$$\begin{aligned} \Theta _{L}^j = e^{- \sqrt{{\widetilde{A}^j_1}/{\varepsilon }} \, z}, \qquad \Theta _{R}^j = e^{- \sqrt{{\widetilde{A}^j_N}/{\varepsilon }} \, (1-z)}. \end{aligned}$$
(3.35)

Note that it is also possible to solve for and hence use \(\theta ^j_{2, *}\), \(*=L, R\), whose expressions are more complicated than those of \(\Theta _{*}^j\). However, enriching the cFV by using \(\theta ^j_{2, *}\) instead of \(\Theta _{*}^j\) does not improve the quality of our enriched scheme, especially when the mesh size is bigger than the size of boundary layers, i.e., \(h \ge \sqrt{\varepsilon }\). The effect of the right-hand side of (3.31) is well-reflected in our eFV by (3.40) below.

Now, to construct the enriched approximation of \({u}_{2}^j\), solution of (3.15), we define the eFV space \(\widetilde{V}_{2, h}^j\) by adding into the classical FV space \(V_h\) the basis functions \( \Theta _{L}^j \) and \( \Theta _{R}^j \):

$$\begin{aligned} \widetilde{V}_{2, h}^j = \left\{ \begin{array}{l} \displaystyle \widetilde{u}_{2,h}^j = \sum _{k= 0}^{N+1} \widetilde{u}_{k}^j \chi _{I_k} + r_L^j \Theta _{L}^j (z) + r_R^j \Theta _{R}^j (z) , \, \, \widetilde{u}_{k}^j, \, r_L^j, \, r_R^j \in \mathbb {R}, \\ \text {such that } \,\, { (\widetilde{u}_{0}^j + \widetilde{u}_{1}^j) + 2 r_{L}^j = 0, \, \, \, (\widetilde{u}_{N+1}^j + \widetilde{u}_{N}^j) + 2 r_{R}^j = 0 . } \end{array}\right\} . \end{aligned}$$
(3.36)

Then, we consider the \(L^2\) inner products of the Eq. (3.15) against the basis functions \(\Theta _{L}^j\), \(\Theta _{R}^j\), and \(\chi _{I_{k}}\), \(1 \le k \le N\):

(i):

We multiply the Eq. (3.15)\(_1\) by \(\Theta _{L}^j\) and integrate over \(I_{1} = (0, \, z_{3/2})\). Integrating the diffusive term by parts, we obtain,

$$\begin{aligned} \int _{I_1} \widetilde{A}^j_1 u^{j}_2 \Theta _{L}^j dz {-}\varepsilon \Big ( (d_z u^{j}_2 \Theta _{L}^j) (z_{ \frac{3}{2}}) - (d_z u^{j}_2 \Theta _{L}^j) (0) \Big ) {+}\varepsilon \int _{I_1} d_z u^{j}_2 d_z \Theta _{L}^j dz{=}\!\int _{I_1} f^j_2 \Theta _{L}^j dz. \end{aligned}$$
(3.37)
(ii):

Multiplying (3.15)\(_2\) by \(\Theta _{R}^j\), integrating over \(I_{N} = (z_{N - 1/2}, \, 1)\), we similarly obtain,

$$\begin{aligned}&\int _{I_N} \widetilde{A}^j_N u^{j}_2 \Theta _{R}^j \, dz -\varepsilon \Big ((d_z u^{j}_2 \Theta _{R}^j) (1) - (d_z u^{j}_2 \Theta _{R}^j) (z_{N-\frac{1}{2}}) \Big )\nonumber \\&\quad + \varepsilon \int _{I_N} d_z u^{j}_2 d_z \Theta _{R}^j \, dz = \int _{I_N} f^j_2 \Theta _{R}^j \, dz. \end{aligned}$$
(3.38)
(iii):

Multiplying (3.15)\(_2\) by \(\chi _{I_k}\), \(1 \le k \le N\), integrating over \(\Omega _z = (0, \, 1)\), and integrating the diffusive term by parts, we find that

$$\begin{aligned} \int _{I_k} \widetilde{A}^j_k u^{j}_2 \, dz -\varepsilon \Big ( d_z u^{j}_2 (z_{k +\frac{1}{2}}) - d_z u^{j}_2 (z_{k - \frac{1}{2}}) \Big ) =\int _{I_k} f^j_2 \, dz, \qquad 1 \le k \le N. \end{aligned}$$
(3.39)

Because the \({\Theta }^j_{* }\), \(* = L, R\), satisfy (3.34), we first notice that the projections of the Eqs. (3.37)–(3.39) into the enriched space \(\widetilde{V}^j_{2, h}\) are the same as those with \(u^{j}_2\) replaced by the step function \(\sum _{k= 0}^{N+1} \widetilde{u}^j_{k} \chi _{I_k}\). Then, in order to take into account the right-hand side of (3.31), we additionally include in the scheme the extra terms,

$$\begin{aligned} \int _{I_1} - i (2 \pi j /L) r_{L} e^{-z/\sqrt{\varepsilon }} (u^0_2)^j \, \Theta ^j_L \, dz \, r^{j}_L, \quad \int _{I_N} - i (2 \pi j /L) r_{R} e^{-(1-z)/\sqrt{\varepsilon }} (u^0_2)^j \, \Theta ^j_L \, dz \, r^{j}_R, \end{aligned}$$
(3.40)

which are originated from the interaction of the non-linear terms \(u^{\varepsilon }_1 \, {\partial _x u^{\varepsilon }_2}\) and \(u^{0}_1 \, {\partial _x u^{0}_2}\).

Using (3.23) for \(\widetilde{V}^j_{2, h}\) as well, we finally obtain the following projections of (3.37)–(3.39) onto the enriched space \(\widetilde{V}^j_{2, h}\):

Find\(\widetilde{u}^j_{2, h} \in \widetilde{V}^j_{2,h}\)such that

$$\begin{aligned}&\Big ( i (2 \pi j / L) r_{L} \int _{I_1} e^{-z/\sqrt{\varepsilon }} (u^0_2)^j \Theta ^j_L \, dz \Big ) r^j_L\nonumber \\&\quad -\dfrac{\varepsilon }{h_{\frac{1}{2}}} \Big ( 1 + \dfrac{1}{2} \int _{I_1} d_z \Theta ^j_L \, dz \Big ) \widetilde{u}_{0}^j\nonumber \\&\quad + \bigg [ \widetilde{A}^j_1 \int _{I_1} \Theta ^j_L \, dz +\dfrac{\varepsilon }{h_{\frac{3}{2}}} \Theta ^j_L (z_{\frac{3}{2}}) +\dfrac{\varepsilon }{h_{\frac{1}{2}}} -\dfrac{\varepsilon }{2} \Big ( \dfrac{1}{h_{\frac{3}{2}}} - \dfrac{1}{h_{\frac{1}{2}}} \Big ) \int _{I_1} d_z \Theta ^j_L \, dz\bigg ] \widetilde{u}_{1}^j \nonumber \\&\quad -\dfrac{\varepsilon }{h_{\frac{3}{2}}} \Big ( \Theta ^j_L(z_{\frac{3}{2}}) - \dfrac{1}{2} \int _{I_1} d_z \Theta ^j_L \, dz \Big ) \widetilde{u}_{2}^j = f_2^j(z_1) \int _{I_1} \Theta ^j_L \, dz, \end{aligned}$$
(3.41)
$$\begin{aligned}&-\dfrac{\varepsilon }{h_{k- \frac{1}{2}}} \widetilde{u}_{k-1}^j +\bigg (\widetilde{A}^j_k h_k + \dfrac{\varepsilon }{h_{k- \frac{1}{2}}} +\dfrac{\varepsilon }{h_{k+ \frac{1}{2}}} \bigg ) \widetilde{u}_{k}^j -\dfrac{\varepsilon }{h_{k+ \frac{1}{2}}} \widetilde{u}_{k+1}^j = f_2^j(z_k) h_k, \quad 1 \le k \le N,\nonumber \\ \end{aligned}$$
(3.42)

and

$$\begin{aligned}&\Big ( i (2 \pi j / L) r_{R} \int _{I_N} e^{-(1-z)/\sqrt{\varepsilon }} (u^0_2)^j \Theta ^j_R \, dz \Big ) r^j_R\nonumber \\&\quad -\dfrac{\varepsilon }{h_{N-\frac{1}{2}}} \Big ( \Theta ^j_R (z_{N-\frac{1}{2}}) + \dfrac{1}{2} \int _{I_N} d_z \Theta ^j_R \, dz \Big ) \widetilde{u}_{N-1}^j \nonumber \\&\quad + \bigg [ \widetilde{A}^j_N \int _{I_N} \Theta ^j_R \, dz +\dfrac{\varepsilon }{h_{N+\frac{1}{2}}} +\dfrac{\varepsilon }{h_{N-\frac{1}{2}}} \Theta ^j_R (z_{N-\frac{1}{2}}) -\dfrac{\varepsilon }{2} \Big (\dfrac{1}{h_{N+\frac{1}{2}}} - \dfrac{1}{h_{N-\frac{1}{2}}} \Big ) \int _{I_N} d_z \Theta ^j_R \, dz\bigg ] \widetilde{u}_{N}^j\nonumber \\&\quad -\dfrac{\varepsilon }{h_{N+\frac{1}{2}}} \Big ( 1 - \dfrac{1}{2} \int _{I_N} d_z \Theta ^j_R \, dz \Big ) \widetilde{u}_{N+1}^j =f_2^j(z_N) \int _{I_N} \Theta ^j_R \, dz. \end{aligned}$$
(3.43)

Solving the equations above, we obtain the eFV approximation \(\widetilde{u}_{2, h}^j\) for \(-N_x \le j \le N_x\). Then the eFV solution \(\widetilde{u}_{2, h}\) of \(u_{2}^{\varepsilon }\) is obtained by

$$\begin{aligned} \widetilde{u}_{2, h} = \sum _{|j| \le N_x} \widetilde{u}_{2, h}^j e^{2 \pi j i x / L}. \end{aligned}$$
(3.44)

Comparison of the Classical FV (cFV) and Enriched FV (eFV) Approximations for the Velocity

To construct an explicit example for numerical computations, we choose the data \(f_1\) as \( f_1(z) = 1 + z(1-z) \), and find the exact solution \(u^{\varepsilon }_1\) of (1.2) and the corresponding limit solution \(u^0_1\) of (1.5) in the form,

$$\begin{aligned} u^{\varepsilon }_1 (z) = (1 - 2 \varepsilon ) \bigg ( 1 - \dfrac{1 -e^{-\frac{1}{\sqrt{\varepsilon }}}}{1 - e^{-\frac{2}{\sqrt{\varepsilon }}}}\Big (e^{-\frac{z}{\sqrt{\varepsilon }}} +e^{-\frac{1-z}{\sqrt{\varepsilon }}} \Big ) \bigg ) + z(1-z), \qquad u^{0}_1 (z) = 1+ z(1-z). \end{aligned}$$
(3.45)

By sequentially choosing the data \(f_2\) as

$$\begin{aligned} f_2(x, z) = f_1(z)(1 + \sin (2 \pi x)) + 4 \pi ^2 \varepsilon \, u^{\varepsilon }_1(z)\sin (2 \pi x) + 2 \pi (u^{\varepsilon }_1(z))^2 \cos (2 \pi x), \end{aligned}$$

we find the exact solution \(u^{\varepsilon }_2\) of (1.2) with \(L=1\) and the corresponding limit solution \(u^0_2\) of (1.5) with \(L=1\) in the form,

$$\begin{aligned} u^{\varepsilon }_2 (x, z) = u^{\varepsilon }_1 (z) (1 + \sin (2 \pi x)), \qquad u^{0}_2 (x, z) = \big (1 + z(1-z) \big ) ( 1 + \sin (2 \pi x)). \end{aligned}$$
(3.46)

Here we see the mismatch of boundary values because \(u^\varepsilon _1 =u^\varepsilon _2 = 0\) at \(z = 0, 1\) while \(u^0_1 = 1\) and \(u^0_2 = \sin (2 \pi x)\) at \(z = 0, 1\), and hence the boundary layers appear near the boundary at \(z = 0,1\); see the graphs of \(u^\varepsilon _1\) and \(u^\varepsilon _2\) in Fig. 1 below when \(\varepsilon = 10^{-5}\).

Fig. 1
figure1

Exact solutions \(u^{\varepsilon }_1\) (left) and \(u^{\varepsilon }_2\) (right) when \(\varepsilon = 10^{-5}\)

To demonstrate how the boundary layers effect on the classical method and numerically verify how much our new enriched method improve the performance, we compute the both cFV and eFV approximation of the example above for a uniform mesh of size \(h=1/100\), but with different values of the viscosity as \(\varepsilon =10^{-m}\), \(3 \le m \le 8\) below. As boundary layers are the highly local phenomena near the boundary, in order to measure the computational error, we use the normalized relative\(L^\infty \)error, not any type of averaging norms, e.g., \(L^2\) or relative \(L^2\).

As we can see in Figs. 2 and 3, and Tables 1 and 2, the dominant computational error for cFV appear near the boundary by the effect of boundary layers, and it is significantly resolved by our new eFV.

Fig. 2
figure2

Comparison of cFV and eFV approximations for \(u^{\varepsilon }_1\) when \(N=100\) and \(\varepsilon = 10^{-7}\). \(\hbox {cFV error}=\mathbf 0.0020 \) and \(\hbox {eFV error}=\mathbf 1.9999e-07 \)

Fig. 3
figure3

Comparison of cFV and eFV approximations for \(u^{\varepsilon }_2\) when \(N=100\) and \(\varepsilon = 10^{-7}\). \(\hbox {cFV error}=0.0040\) and \(\hbox {eFV error}=3.1293e-04\)

For a fixed viscosity parameter \(\varepsilon = 10^{-6}\), we compute and compare the relative \(L^{\infty }\) error of cFV and eFV approximations as in Table 3. It appears clear that our new eFV scheme produce much better numerical result than the classical one. Moreover, the numerical result for \(u^\varepsilon _2\), shown in Table 3, emphasizes the comparison between the cFV error and eFV error, especially when the stiff boundary layer of the problem affects on the classical cFV scheme. Here, when \(\varepsilon = 10^{-6}\), the size of boundary layer is \(10^{-3}\), and in fact, we notice from Table 3 that the cFV error of \(u^\varepsilon _2\) is relatively large when the mesh size 1/N is comparable with the size boundary layer, i.e., when \(300 \le N \le 1000\). We also notice that our new eFV scheme improves the quality of numerical approximations for \(u^\varepsilon _2\) in this problematic range of mesh size. As the mesh size is getting further smaller \((1/N \le 10^{-3})\), both cFV error and eFV error decrease.

Table 1 Relative \(L^{\infty }\) error of cFV and eFV approximations for \(u^{\varepsilon }_1\) when \(N=100\)
Table 2 Relative \(L^{\infty }\) error of cFV and eFV approximations for \(u^{\varepsilon }_2\) when \(N=100\)
Table 3 Relative \(L^{\infty }\) error of cFV and eFV solutions for \(\varvec{u}^{\varepsilon }\) when \(\varepsilon =10^{-6}\)

FV Approximations of the Vorticity Vector Field

We construct the Finite Volume (FV) approximations of the vorticity,

$$\begin{aligned} \varvec{\omega }^{\varepsilon } = ( \omega ^\varepsilon _1(x, z), \omega ^\varepsilon _2(z), \omega ^\varepsilon _3(x, z)) := \text {curl }\varvec{u}^{\varepsilon } = \big ( -\partial _z u^\varepsilon _2(x, z), \, d_z u^\varepsilon _1,(z)\, \partial _x u^\varepsilon _2 (x, z)\big ), \end{aligned}$$
(4.1)

which satisfies the Eqs. (1.7)–(1.8). Note that the equation for the third component \(\omega ^\varepsilon _3 \) is identical to that of \(u^{\varepsilon }_2\) (with different data), whose classical FV (cFV) and enriched FV (eFV) approximations are well-studied in the previous Sect. 3. Hence, here we focus on computations for the first and second components \(\omega ^\varepsilon _1\) and \(\omega ^\varepsilon _2\) only.

In Sect. 4.1, following the methodology introduced in Sect. 3, we construct the cFV approximations of \(\varvec{\omega }^\varepsilon \) when the viscosity parameter is small. We will observe below that the dominant error occurs near the boundary at \(z = 0, 1\), especially when the viscosity \(\varepsilon \) is relatively small with respect to the mesh size h, i.e., \(\varepsilon ^{1/2} < h\). In Sect. 4.2, we enrich the classical FV scheme by adding the curl of the corrector, used in Sect. 3, i.e., \(d_z \Theta ^j_*\) for \(\omega ^\varepsilon _1\) and \(d_z \overline{\theta }_{1, *}\) for \(\omega ^\varepsilon _2\), \(*=L,R\). Performing numerical computations for both cFV and eFV schemes, we verify below that our novel enriched scheme (eFV) reduces significantly the computational error near the boundary \(\Gamma \). The computational error of the cFV and eFV schemes for \(\omega ^\varepsilon _1\) and \(\omega ^\varepsilon _2\) are compared in Sect. 4.3.

Classical FV (cFV) Approximations of the Vorticity Vector Field

Concerning the cFV approximations of \(\varvec{\omega }^\varepsilon \), we first consider the simple component \(\omega ^\varepsilon _2\).

Recalling the 1D FV setting introduced in Sect. 4, we define a FV space for \(\omega ^\varepsilon _2\) by

$$\begin{aligned} W_{2,h} = \Big \{ w_{2,h} = \sum _{k= 0}^{N+1} w_k \chi _{I_k}, \Big | \, { w_0 = w_1 + \dfrac{h_{1/2}}{\varepsilon } f_1(0), \, \, \, w_{N+1} = w_N - \dfrac{h_{N+1/2}}{\varepsilon } f_1(1) } \Big \}. \end{aligned}$$
(4.2)

The constraints in the definition of \(W_{2,h}\) enforce the boundary condition (1.8)\(_2\) on the cFV approximation \(\widetilde{w}_{2,h}\), i.e., at \(z = 0\),

$$\begin{aligned} \nabla _h w_{2, h}|_{z= 0} = \dfrac{w_{ 1} - w_{ 0}}{h_{1/2}} = - \dfrac{1}{\varepsilon } f_1(0). \end{aligned}$$
(4.3)

Now, applying the 1D cFV discretization for (1.7)\(_2\), as we did in Sect. 3.1, we obtain the cFV approximation \(w_{2, h} \in W_{2,h}\) for the solution \(\omega ^{\varepsilon }_2\) to (1.7)\(_2\) - (1.8)\(_2\).

The third component \(\omega ^\varepsilon _3 \) satisfies the equation [see (1.7)–(1.8)] identical to that of \(u^{\varepsilon }_2\), but with a different external force \(\partial _x f_2\). Hence applying the exactly same process as for \(u^{\varepsilon }_2\), we compute the cFV approximation \(w_{3, h}\) of \(\omega ^{\varepsilon }_3\) in the form,

$$\begin{aligned} w_{3, h} = \sum _{|j| \le N_x} {w}_{3, h}^j e^{2 \pi j i x / L}, \end{aligned}$$
(4.4)

where \({w}_{3, h}^j \in V_h\) is the 1D cFV approximation of solution \({w}_{3}^j\) to

$$\begin{aligned} \left\{ \begin{array}{l} \Big (1 + \varepsilon (2 \pi j / L)^2 + i (2 \pi j / L) u^{\varepsilon }_{1} \Big ){w}_{3}^j - \varepsilon d_z^2 {w}_{3}^j = \partial _x {f}_{2}^{j}, \quad 0< z < 1,\\ {w}_{3}^j = 0, \quad z = 0, 1. \end{array}\right. \end{aligned}$$
(4.5)

Here \(\partial _x {f}_{2}^{j} = \partial _x {f}_{2}^{j}(z)\) is the j-th Fourier coefficient of \(\partial _x f_2\), and use the cFV solution \(u_{1, h} = \sum _{k=0}^{N+1} u_{k} \chi _{I_k}\) to approximate \(u^{\varepsilon }_{1}\).

For the first component \(\omega ^{\varepsilon }_1 = \omega ^{\varepsilon }_1(x, z)\), taking the partial Fourier expansion in x, up to the mode \(N_x>0\), and write

$$\begin{aligned} \omega ^{\varepsilon }_1 (x, z) \cong \sum _{|j| \le N_x} w_{1}^j(z) e^{2 \pi j i x / L}. \end{aligned}$$
(4.6)

Then, by taking the Fourier transform in x of (1.7)–(1.8), we write the equation of \(\omega ^{\varepsilon }_{1}\) mode by mode: For \(-N_x \le j \le N_x\),

$$\begin{aligned} \left\{ \begin{array}{l} \Big (1 + \varepsilon (2 \pi j / L)^2 + i (2 \pi j / L) u^{\varepsilon }_{1} \Big ){w}_{1}^j - \varepsilon d_z^2 {w}_{1}^j = - \partial _z{f}_{2}^{j} + w_{2, h} \, w_{3}^j, \quad 0< z < 1,\\ \partial _z {w}_{1}^j = \dfrac{1}{\varepsilon } f^j_2, \quad z = 0, 1, \end{array}\right. \end{aligned}$$
(4.7)

where \({f}_{2}^{j} = {f}_{2}^{j}(z)\) and \(\partial _z {f}_{2}^{j} = \partial _z {f}_{2}^{j}(z)\) are the j-th Fourier coefficient of \(f_2\) and \(\partial _z f_2\).

To enforce the boundary condition (4.7)\(_2\) into the FV space, we define

$$\begin{aligned} W_{1,h}^j = \Big \{ w_{1, h}^j = \sum _{k= 0}^{N+1} w_k^j \chi _{I_k}, \Big | \, { w_0^j = w_1^j - \dfrac{h_{1/2}}{\varepsilon } f_2^j(0), \, \, \, w_{N+1}^j = w_N^j + \dfrac{h_{N+1/2}}{\varepsilon } f_2^j(1) } \Big \}. \end{aligned}$$
(4.8)

Now, applying the 1D cFV discretization to (4.7), with using the cFV approximations \(u_{1, h}\), \(w_{2, h}\) and \(w_{3, h}^j\) of \(u^{\varepsilon }_1\), \(\omega ^{\varepsilon }_2\), and \(w^{j}_3\), which are already computed in advance, we obtain the cFV approximation \({w}_{1, h}^{j} \in W_{1,h}^j\) of solution \({w}_{1}^{j}\) to (4.7). Then the cFV solution \(w_{1, h}\) of \(\omega _{1}^{\varepsilon }\) is obtained by

$$\begin{aligned} w_{1, h} = \sum _{|j| \le N_x} {w}_{1, h}^j e^{2 \pi j i x / L}. \end{aligned}$$
(4.9)

In conclusion, we have constructed the classical FV (cFV) solutions \(w_{2, h}\), \(w_{3, h}\), and \(w_{1, h}\) in a sequel, which approximates the vorticity vector field \(\varvec{\omega }^{\varepsilon }\) satisfying (1.7)–(1.8).

Enriched FV (eFV) Approximations of the Vorticity Vector Field

We enrich the classical FV schemes for each component of the vorticity by adding the curl of the corrector that we defined and used in Sect. 3. As it appears below, we use \(d_z \Theta ^j_*\) for \(\omega ^\varepsilon _1\) and \(d_z \overline{\theta }_{1, *}\) for \(\omega ^\varepsilon _2\), \(*=L,R\).

For the component \(\omega ^\varepsilon _2\), adding the correctors,

$$\begin{aligned} d_z \overline{\theta }_{1, L} = - \dfrac{1}{\sqrt{\varepsilon }} e^{-z/\sqrt{\varepsilon }}, \qquad \qquad d_z \overline{\theta }_{1, R} =\dfrac{1}{\sqrt{\varepsilon }} e^{-(1-z)/\sqrt{\varepsilon }}, \end{aligned}$$
(4.10)

we define the enriched FV space for \(\omega ^\varepsilon _2\) as

$$\begin{aligned} \widetilde{W}_{2,h} = \left\{ \begin{array}{l} \displaystyle \widetilde{w}_{2,h} = \sum _{k= 0}^{N+1} \widetilde{w}_{k} \chi _{I_k} + r_L d_z \overline{\theta }_{1, \, L} (z) + r_R d_z \overline{\theta }_{1, \, R} (z) , \, \, \widetilde{w}_{k}, \, r_L, \, r_R \in \mathbb {R}, \\ \text {such that } \,\, { \dfrac{\widetilde{w}_1 -\widetilde{w}_0}{h_{1/2}} + \dfrac{r_L}{{\varepsilon }} =-\dfrac{f_1(0)}{\varepsilon }, \, \, \, \dfrac{\widetilde{w}_{N+1} -\widetilde{w}_N}{h_{N+1/2}} + \dfrac{r_R}{{\varepsilon }} =-\dfrac{f_1(1)}{\varepsilon } } \end{array}\right\} . \end{aligned}$$
(4.11)

Note that the constraints in the definition of \(\widetilde{W}_{2,h}\) enforce the boundary condition (1.8)\(_2\) on the eFV approximation \(\widetilde{w}_{2,h}\). Then, applying the 1D cFV discretization for (1.7)\(_2\), as we did in Sect. 3.2, we obtain the eFV approximation \(\widetilde{w}_{2, h} \in \widetilde{W}_{2,h}\) for the solution \(\omega ^{\varepsilon }_2\) to (1.7)\(_2\)–(1.8)\(_2\), which satisfies the discrete equations identical to (3.27)–(3.28) with \(f_1\) replaced by \(d_z f_1\) and the enriched space \(\widetilde{V}_{1, h}\) replaced by \(\widetilde{W}_{2, h}\) above.

For the enriched FV (eFV) approximation \(\widetilde{w}_{3, h}\) of \(\omega ^{\varepsilon }_3\) in the form,

$$\begin{aligned} \widetilde{w}_{3, h} = \sum _{|j| \le N_x} \widetilde{w}_{3, h}^j e^{2 \pi j i x / L}, \end{aligned}$$
(4.12)

we follow exactly the same process applied for \(\widetilde{u}^j_{2, h}\) in (3.41) (but with a different external force \(\partial _x f_2\) instead of \(f_2\), and compute \(\widetilde{w}_{3, h}^j \in \widetilde{V}^j_{2, h}\) as the eFV approximation of solution \({w}_{3}^j\) to (4.5).

Now, to construct the enriched FV approximation of the first component \(\omega ^{\varepsilon }_1 = \omega ^{\varepsilon }_1(x, z)\), we recall the Fourier expansion in x, up to the mode \(N_x>0\),

$$\begin{aligned} \omega ^{\varepsilon }_1 (x, z) \cong \sum _{|j| \le N_x} w_{1}^j(z) e^{2 \pi j i x / L}. \end{aligned}$$
(4.13)

where each \({w}_{1}^j\) is the solution of (4.7).

Recalling from (3.35), we differentiate \(\Theta _{*}^j\), \(*=L, R\), in the z variable, and introduce the correctors for \(\omega ^{\varepsilon }_1\) as

$$\begin{aligned} d_z \Theta _{L}^j = - \sqrt{\dfrac{\widetilde{A}^j_1}{\varepsilon }} \, e^{- \sqrt{{\widetilde{A}^j_1}/{\varepsilon }} \, z}, \qquad d_z \Theta _{R}^j = \sqrt{\dfrac{\widetilde{A}^j_N}{\varepsilon }} \, e^{- \sqrt{{\widetilde{A}^j_N}/{\varepsilon }} \, (1-z)}. \end{aligned}$$
(4.14)

Then, by enforcing the boundary condition (4.7)\(_1\) into the eFV space, we define

$$\begin{aligned} \widetilde{W}_{1,h}^j = \left\{ \begin{array}{l} \displaystyle \widetilde{w}_{1,h}^j = \sum _{k= 0}^{N+1} \widetilde{w}_{k}^j \chi _{I_k} + r_L^j d_z \Theta _{L}^j (z) + r_R^j d_z \Theta _{R}^j (z) , \, \, \widetilde{w}_{k}^j, \, r_L^j, \, r_R^j \in \mathbb {R}, \\ \text {such that } \,\, { \dfrac{\widetilde{w}_1^j -\widetilde{w}_0^j}{h_{1/2}} + \dfrac{\widetilde{A}^j_1}{{\varepsilon }}r_L^j = \dfrac{f_2^j(0)}{\varepsilon }, \, \, \, \dfrac{\widetilde{w}_{N+1}^j -\widetilde{w}_N^j}{h_{N+1/2}} + \dfrac{\widetilde{A}^j_N}{{\varepsilon }} r_R = \dfrac{f_2^j(1)}{\varepsilon } } \end{array}\right\} . \end{aligned}$$
(4.15)

Now, applying the 1D eFV discretization, and using the eFV approximations \(\widetilde{u}_{1, h}\), \(\widetilde{w}_{2, h}\) and \(\widetilde{w}_{3, h}^j\) of \(u^{\varepsilon }_1\), \(\omega ^{\varepsilon }_2\), and \(w^{j}_3\), which are already computed in advance, we obtain the eFV approximation \(\widetilde{w}_{1, h}^{j} \in \widetilde{W}_{1,h}^j\) of solution \({w}_{1}^{j}\) to (4.7) and hence the eFV solution \(\widetilde{w}_{1, h}\) of \(\omega _{1}^{\varepsilon }\) defined by

$$\begin{aligned} \widetilde{w}_{1, h} = \sum _{|j| \le N_x} \widetilde{w}_{1, h}^j e^{2 \pi j i x / L}. \end{aligned}$$
(4.16)

Comparison of the Classical FV (cFV) and Enriched FV (eFV) Approximations for the Vorticity

By taking curl of the exact solution of \(\varvec{u}^{\varepsilon }\), defined in (3.45) and (3.46), we prepare the exact solution of \(\varvec{\omega }^{\varepsilon }\),

$$\begin{aligned} \left\{ \begin{array}{l} \omega ^{\varepsilon }_1 (x,z) = -\omega ^{\varepsilon }_2(z) (1 + \sin (2 \pi x)),\\ \omega ^{\varepsilon }_2 (z) = - (1 - 2 \varepsilon ) \dfrac{1 -e^{-\frac{1}{\sqrt{\varepsilon }}}}{1 - e^{-\frac{2}{\sqrt{\varepsilon }}}} \Big (\dfrac{-1}{\sqrt{\varepsilon }}e^{-\frac{z}{\sqrt{\varepsilon }}} +\dfrac{1}{\sqrt{\varepsilon }} e^{-\frac{1-z}{\sqrt{\varepsilon }}} \Big )+ 1-2z, \\ \omega ^{\varepsilon }_3 (z) = 2 \pi \, u^{\varepsilon }_1(z) \cos (2 \pi x). \end{array}\right. \end{aligned}$$
(4.17)

As mentioned before, the equation for \(\omega ^\varepsilon _3 \) is identical to that of \(v^{\varepsilon }_2\), whose cFV and eFV approximations are well-studied in the previous Sect. 3. Hence, here present the numerical results of \(\omega ^\varepsilon _1\) and \(\omega ^\varepsilon _2\) only.

We notice that, at the vanishing viscosity limit \(\varepsilon =0\), the \(\omega ^{\varepsilon }_i\), \(i=1,2\), blows up near the boundary at \(z=0,1\) as fast as the approximation of identity, e.g., \(1/\sqrt{\varepsilon } e^{-z/\sqrt{\varepsilon }}\) near \(z=0\); see the graphs of \(\omega ^\varepsilon _1\) and \(\omega ^\varepsilon _2\) in Fig. 4 below where they change very stiff near the boundary at \(z = 0,1\).

Fig. 4
figure4

Exact solutions \(\omega ^{\varepsilon }_1\) (left) and \(\omega ^{\varepsilon }_2\) (right) when \(\varepsilon = 10^{-5}\)

To demonstrate the effect of boundary layers and show how much our new enriched method improve the performance, we compute the both cFV and eFV approximation of the example above for a uniform mesh of size \(h = 1/100\), but with different values of the viscosity, \(\varepsilon =10^{-m}\), \(4 \le m \le 9\) below.

In the case when \(\varepsilon \le 10^{-7}\) and \(N=100\), the boundary layer near \(z = 0\) is very far from the center \(z = 1/(2N)\) of the first control volume, while the value of \(\omega ^\varepsilon _1\) or \(\omega ^\varepsilon _2\) inside the boundary layer near \(z = 0\) is as large as \(\varepsilon ^{-1} \, e^{-z/\varepsilon }\). Hence, as we notice from Tables 4 and 5, the computational error of the cFV (approximating \(\omega ^\varepsilon _1\) or \(\omega ^\varepsilon _2\) at \(z = 1/(2N)\)) is very large when \(\varepsilon \le 10^{-7}\). Our enriched FV scheme naturally captures this extremely singular behavior by using the corrector and hence reduces the computational error significantly.

Fig. 5
figure5

Comparison of cFV and eFV solutions for \(\omega ^{\varepsilon }_1\) when \(N=100\) and \(\varepsilon = 10^{-7}\). \(\hbox {cFV error}=199.7956\) and \(\hbox {eFV error}=1.1888e-04\)

Fig. 6
figure6

Comparison of cFV and eFV approximations for \(\omega ^{\varepsilon }_2\) when \(N=100\) and \(\varepsilon = 10^{-7}\). \(\hbox {cFV error}=99.8997\) and \(\hbox {eFV error}=3.7560e-05\)

Table 4 Relative \(L^{\infty }\) error of cFV and eFV approximations for \(\omega ^{\varepsilon }_1\) when \(N=100\)
Table 5 Relative \(L^{\infty }\) error of cFV and eFV approximations for \(\omega ^{\varepsilon }_2\) when \(N=100\)

As we can see in Figs. 5 and 6, and Tables 4 and 5, the dominant computational error for cFV appear near the boundary by the effect of boundary layers, and it is significantly resolved by our new eFV.

For a fixed viscosity parameter \(\varepsilon = 10^{-6}\), we compute and compare the relative \(L^{\infty }\) error of cFV and eFV approximations as in Table 6 below. It appears clear that our new eFV scheme produce much better numerical result than the classical one.

Table 6 Relative \(L^{\infty }\) error of cFV and eFV solutions for \(\omega ^{\varepsilon }_1\) and \(\omega ^{\varepsilon }_2\) when \(\varepsilon =10^{-6}\)

Conclusion

In this work, we have presented a semi-analytic approach to improve numerical quality of the classical Finite Volume (cFV) method, applied to singularly perturbed non-linear fluid equations. Invoking the singular perturbation analysis, we first derived the so-called corrector which is an analytic approximation of the velocity vector field near the boundary. Our rigorous boundary layer analysis confirms that the stiffness of solutions to the singular perturbation problem is then resolved and captured by the corrector. By embedding this corrector into the cFV schemes, we construct our new enriched FV (eFV) schemes for the velocity and vorticity equations, and numerically verify that our novel eFV schemes reduce significantly the computational error of cFV schemes (especially near the boundary), and hence produce better approximations.

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Acknowledgements

The first author was supported partially by Collaboration Grant for Mathematicians, Simons Foundation and Research - RII Grant, Office of the Executive Vice President for Research and Innovation, University of Louisville. The second and third authors were supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education (2018R1D1A1B07048325) and the Research Fund (1.190136.01) of UNIST.

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Gie, G., Jung, C. & Lee, H. Enriched Finite Volume Approximations of the Plane-Parallel Flow at a Small Viscosity. J Sci Comput 84, 7 (2020). https://doi.org/10.1007/s10915-020-01259-0

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Keywords

  • Finite Volume method
  • Plane-parallel flow
  • Navier–Stokes equations
  • Boundary layers