Tutorial on Hybridizable Discontinuous Galerkin (HDG) Formulation for Incompressible Flow Problems

Abstract

A hybridizable discontinuous Galerkin (HDG) formulation of the linearized incompressible Navier-Stokes equations, known as Oseen equations, is presented. The Cauchy stress formulation is considered and the symmetry of the stress tensor and the mixed variable, namely the scaled strain-rate tensor, is enforced pointwise via Voigt notation. Using equal-order polynomial approximations of degree k for all variables, HDG provides a stable discretization. Moreover, owing to Voigt notation, optimal convergence of order $k+1$ is obtained for velocity, pressure and strain-rate tensor and a local postprocessing strategy is devised to construct an approximation of the velocity superconverging with order $k+2$, even for low-order polynomial approximations. A tutorial for the numerical solution of incompressible flow problems using HDG is presented, with special emphasis on the technical details required for its implementation.

Appendices

Appendix: Saddle-Point Structure of the Global Problem

In this Appendix, the symmetry of the global system in (32) is demonstrated. First, rewrite (32) as

$$\begin{aligned} \begin{bmatrix}\widehat{\mathbf {K}} &{} \mathbf {H} \\ \mathbf {G}^T &{} \mathbf {0} \end{bmatrix} \begin{Bmatrix}{\hat{\mathbf {u}}} \\ \varvec{\rho } \end{Bmatrix} = \begin{Bmatrix}{\hat{\mathbf {f}}}_{\hat{u}} \\ {\hat{\mathbf {f}}}_{\rho } \end{Bmatrix}, \end{aligned}$$

(43)

where the block $\mathbf {H}$ is obtained by the solution of the local problem in (30) and has the following form

(44)

In order for the system in (43) to have a saddle-point structure, it needs to be proved that $\mathbf {H} = \mathbf {G}$. For the sake of readability, rewrite the matrix of the local problem in (30) using the block structure

$$\begin{aligned} \mathbf {K}_{\! e}:= \begin{bmatrix} \mathbf {B}_{\! e}&{} \mathbf {C}_{\! e}\\ \mathbf {C}_{\! e}^T &{} \mathbf {D}_{\! e}\end{bmatrix} \end{aligned}$$

(45)

where the blocks are defined as

$$\begin{aligned} \mathbf {B}_{\! e}:= \begin{bmatrix} \mathbf {A}_{LL} &{} \mathbf {A}_{Lu} \\ \mathbf {A}_{Lu}^T &{} \mathbf {A}_{uu} \end{bmatrix} , \quad \mathbf {C}_{\! e}:= \begin{bmatrix} \mathbf {0} &{} \mathbf {0} \\ \mathbf {A}_{pu}^T &{} \mathbf {0} \end{bmatrix} , \quad \mathbf {D}_{\! e}:= \begin{bmatrix} \mathbf {0} &{} \mathbf {a}_{\rho p}^T \\ \mathbf {a}_{\rho p}&{} \mathbf {0} \end{bmatrix} . \end{aligned}$$

Proposition 5.1

For each element $\Omega _e$, it holds

$$\begin{aligned} \mathbf {K}_{\! e}^{-1} \begin{Bmatrix} \mathbf {0} \\ \mathbf {0} \\ \mathbf {0} \\ 1 \end{Bmatrix}_{\! e} = \begin{bmatrix} - \begin{bmatrix} \mathbf {B}_{\! e}^{-1} \end{bmatrix}_{12} \mathbf {A}_{pu}^T \begin{bmatrix} \mathbf {S}_{\! e}^{-1} \end{bmatrix}_{12}\\ - \begin{bmatrix} \mathbf {B}_{\! e}^{-1} \end{bmatrix}_{22} \mathbf {A}_{pu}^T \begin{bmatrix} \mathbf {S}_{\! e}^{-1} \end{bmatrix}_{12}\\ \begin{bmatrix} \mathbf {S}_{\! e}^{-1} \end{bmatrix}_{12}\\ 0 \end{bmatrix}_{\! e} , \end{aligned}$$

(46)

where

$$\begin{aligned} \begin{aligned} \begin{bmatrix} \mathbf {B}_{\! e}^{-1} \end{bmatrix}_{12}&= - \mathbf {A}_{LL}^{-1} \mathbf {A}_{Lu} (\mathbf {A}_{uu} - \mathbf {A}_{Lu}^T \mathbf {A}_{LL}^{-1} \mathbf {A}_{Lu})^{-1} , \\ \begin{bmatrix} \mathbf {B}_{\! e}^{-1} \end{bmatrix}_{22}&= (\mathbf {A}_{uu} - \mathbf {A}_{Lu}^T \mathbf {A}_{LL}^{-1} \mathbf {A}_{Lu})^{-1} , \\ \begin{bmatrix} \mathbf {S}_{\! e}^{-1} \end{bmatrix}_{12}&= \Bigl ( \mathbf {a}_{\rho p} \bigl ( \mathbf {I} - ( \mathbf {A}_{pu} (\mathbf {A}_{uu} - \mathbf {A}_{Lu}^T \mathbf {A}_{LL}^{-1} \mathbf {A}_{Lu})^{-1} \mathbf {A}_{pu}^T )^{\dagger } \\&(\mathbf {A}_{pu} (\mathbf {A}_{uu} - \mathbf {A}_{Lu}^T \mathbf {A}_{LL}^{-1} \mathbf {A}_{Lu})^{-1} \mathbf {A}_{pu}^T ) \bigr ) \Bigr )^{\dagger } . \end{aligned} \end{aligned}$$

Proof

The inverse of the block matrix in (45), written using Schur-Banachiewicz form (see Bernstein 2009, Sect. 2.17), is

$$\begin{aligned} \mathbf {K}_{\! e}^{-1} {:=} \left[ \begin{array}{@{}c@{\;}c@{}} \mathbf {B}_{\! e}^{-1} ( \mathbf {I} + \mathbf {C}_{\! e}(\mathbf {D}_{\! e}-\mathbf {C}_{\! e}^T\mathbf {B}_{\! e}^{-1}\mathbf {C}_{\! e})^{-1}\mathbf {C}_{\! e}^T\mathbf {B}_{\! e}^{-1} ) &{} -\mathbf {B}_{\! e}^{-1}\mathbf {C}_{\! e}(\mathbf {D}_{\! e}-\mathbf {C}_{\! e}^T\mathbf {B}_{\! e}^{-1}\mathbf {C}_{\! e})^{-1} \\ -(\mathbf {D}_{\! e}-\mathbf {C}_{\! e}^T\mathbf {B}_{\! e}^{-1}\mathbf {C}_{\! e})^{-1}\mathbf {C}_{\! e}^T\mathbf {B}_{\! e}^{-1} &{} (\mathbf {D}_{\! e}-\mathbf {C}_{\! e}^T\mathbf {B}_{\! e}^{-1}\mathbf {C}_{\! e})^{-1} \end{array}\right] , \end{aligned}$$

where the block (2, 2) is the inverse of the Schur complement

$$\begin{aligned} \mathbf {S}_{\! e}:= \mathbf {D}_{\! e}-\mathbf {C}_{\! e}^T\mathbf {B}_{\! e}^{-1}\mathbf {C}_{\! e}= \begin{bmatrix} - \mathbf {A}_{pu} \begin{bmatrix} \mathbf {B}_{\! e}^{-1} \end{bmatrix}_{22} \mathbf {A}_{pu}^T &{} \mathbf {a}_{\rho p}^T \\ \mathbf {a}_{\rho p} &{} 0 \end{bmatrix} \end{aligned}$$

(47)

of block $\mathbf {B}_{\! e}$ of the matrix $\mathbf {K}_{\! e}$. Moreover, the block (1, 2) of the inverse matrix $\mathbf {K}_{\! e}^{-1}$ has the form

$$\begin{aligned} \begin{aligned} \begin{bmatrix} \mathbf {K}_{\! e}^{-1} \end{bmatrix}_{12}&= - \mathbf {B}_{\! e}^{-1} \mathbf {C}_{\! e}\mathbf {S}_{\! e}^{-1} \\&= -\begin{bmatrix} \begin{bmatrix} \mathbf {B}_{\! e}^{-1} \end{bmatrix}_{12} \mathbf {A}_{pu}^T \begin{bmatrix} \mathbf {S}_{\! e}^{-1} \end{bmatrix}_{11}&\begin{bmatrix} \mathbf {B}_{\! e}^{-1} \end{bmatrix}_{12} \mathbf {A}_{pu}^T \begin{bmatrix} \mathbf {S}_{\! e}^{-1} \end{bmatrix}_{12} \\ \begin{bmatrix} \mathbf {B}_{\! e}^{-1} \end{bmatrix}_{22} \mathbf {A}_{pu}^T \begin{bmatrix} \mathbf {S}_{\! e}^{-1} \end{bmatrix}_{11}&\begin{bmatrix} \mathbf {B}_{\! e}^{-1} \end{bmatrix}_{22} \mathbf {A}_{pu}^T \begin{bmatrix} \mathbf {S}_{\! e}^{-1} \end{bmatrix}_{12} \end{bmatrix} . \end{aligned} \end{aligned}$$

(48)

To fully determine the inverse matrix $\mathbf {K}_{\! e}^{-1}$, the blocks of $\mathbf {B}_{\! e}^{-1}$ and $\mathbf {S}_{\! e}^{-1}$ need to be computed. Following the Schur-Banachiewicz rationale utilized above, the blocks of $\mathbf {B}_{\! e}^{-1}$ have the form

$$\begin{aligned} \begin{aligned} \begin{bmatrix} \mathbf {B}_{\! e}^{-1} \end{bmatrix}_{11}&:= \mathbf {A}_{LL}^{-1} ( \mathbf {I} + \mathbf {A}_{Lu} (\mathbf {A}_{uu} - \mathbf {A}_{Lu}^T \mathbf {A}_{LL}^{-1} \mathbf {A}_{Lu})^{-1} \mathbf {A}_{Lu}^T \mathbf {A}_{LL}^{-1} ) , \\ \begin{bmatrix} \mathbf {B}_{\! e}^{-1} \end{bmatrix}_{12}&:= - \mathbf {A}_{LL}^{-1} \mathbf {A}_{Lu} (\mathbf {A}_{uu} - \mathbf {A}_{Lu}^T \mathbf {A}_{LL}^{-1} \mathbf {A}_{Lu})^{-1} , \\ \begin{bmatrix} \mathbf {B}_{\! e}^{-1} \end{bmatrix}_{22}&:= (\mathbf {A}_{uu} - \mathbf {A}_{Lu}^T \mathbf {A}_{LL}^{-1} \mathbf {A}_{Lu})^{-1} , \end{aligned} \end{aligned}$$

(49)

and, from the symmetry of $\mathbf {B}_{\! e}$, it follows that $\begin{bmatrix} \mathbf {B}_{\! e}^{-1} \end{bmatrix}_{21} = \begin{bmatrix} \mathbf {B}_{\! e}^{-1} \end{bmatrix}_{12}^T$.

Plugging the expression of $\begin{bmatrix} \mathbf {B}_{\! e}^{-1} \end{bmatrix}_{22}$, see (49), into the definition of $\mathbf {S}_{\! e}$ in (47), it follows that the block (1, 1) of such matrix is

$$\begin{aligned} \begin{bmatrix} \mathbf {S}_{\! e}\end{bmatrix}_{11} := - \mathbf {A}_{pu} (\mathbf {A}_{uu} - \mathbf {A}_{Lu}^T \mathbf {A}_{LL}^{-1} \mathbf {A}_{Lu})^{-1} \mathbf {A}_{pu}^T . \end{aligned}$$

(50)

It is straightforward to observe that this matrix is the Schur complement of block

$$ \begin{bmatrix} \mathbf {A}_{LL} &{} \mathbf {A}_{Lu} \\ \mathbf {A}_{Lu}^T &{} \mathbf {A}_{uu} \end{bmatrix} $$

of the matrix

$$ \begin{bmatrix} \mathbf {A}_{LL} &{} \mathbf {A}_{Lu} &{} \mathbf {0} \\ \mathbf {A}_{Lu}^T &{} \mathbf {A}_{uu} &{} \mathbf {A}_{pu}^T \\ \mathbf {0} &{} \mathbf {A}_{pu} &{} \mathbf {0} \end{bmatrix} , $$

which is singular, since it is obtained from the discretization of an incompressible flow problem with purely Dirichlet boundary conditions. Hence, to compute the blocks of $\mathbf {S}_{\! e}^{-1}$, the framework of the generalized inverse of a partitioned matrix is exploited (see Miao 1991) leading to

$$\begin{aligned} \begin{aligned} \begin{bmatrix} \mathbf {S}_{\! e}^{-1} \end{bmatrix}_{11}&:= \Bigl ( \mathbf {I} - \bigl ( \mathbf {a}_{\rho p} ( \mathbf {I} - \begin{bmatrix} \mathbf {S}_{\! e}\end{bmatrix}_{11}^{\dagger } \begin{bmatrix} \mathbf {S}_{\! e}\end{bmatrix}_{11} ) \bigr ) ^{\dagger } \mathbf {a}_{\rho p} \Bigr ) \\&\begin{bmatrix} \mathbf {S}_{\! e}\end{bmatrix}_{11}^{\dagger } \Bigl ( \mathbf {I} - \mathbf {a}_{\rho p}^T \bigl ( ( \mathbf {I} - \begin{bmatrix} \mathbf {S}_{\! e}\end{bmatrix}_{11} \begin{bmatrix} \mathbf {S}_{\! e}\end{bmatrix}_{11}^{\dagger } ) \mathbf {a}_{\rho p}^T \bigr ) ^{\dagger } \Bigr ) , \\ \begin{bmatrix} \mathbf {S}_{\! e}^{-1} \end{bmatrix}_{12}&:= \bigl ( \mathbf {a}_{\rho p} ( \mathbf {I} - \begin{bmatrix} \mathbf {S}_{\! e}\end{bmatrix}_{11}^{\dagger } \begin{bmatrix} \mathbf {S}_{\! e}\end{bmatrix}_{11} ) \bigr ) ^{\dagger } , \\ \begin{bmatrix} \mathbf {S}_{\! e}^{-1} \end{bmatrix}_{21}&:= \bigl ( ( \mathbf {I} - \begin{bmatrix} \mathbf {S}_{\! e}\end{bmatrix}_{11} \begin{bmatrix} \mathbf {S}_{\! e}\end{bmatrix}_{11}^{\dagger } ) \mathbf {a}_{\rho p}^T \bigr ) ^{\dagger } , \\ \begin{bmatrix} \mathbf {S}_{\! e}^{-1} \end{bmatrix}_{22}&:= 0 \end{aligned} \end{aligned}$$

(51)

where the Moore-Penrose pseudoinverse $\begin{bmatrix} \mathbf {S}_{\! e}\end{bmatrix}_{11}^{\dagger }$ of the singular matrix $\begin{bmatrix} \mathbf {S}_{\! e}\end{bmatrix}_{11}$ has the form

$$\begin{aligned} \begin{bmatrix} \mathbf {S}_{\! e}\end{bmatrix}_{11} ^{\dagger } := - \bigl ( \mathbf {A}_{pu} (\mathbf {A}_{uu} - \mathbf {A}_{Lu}^T \mathbf {A}_{LL}^{-1} \mathbf {A}_{Lu})^{-1} \mathbf {A}_{pu}^T \bigr ) ^{\dagger }. \end{aligned}$$

(52)

From (44), it is straightforward to observe that only the blocks in the last column of the inverse matrix are involved in the definition of the product in (46). The result (46) follows directly from (48), (51) and (52). $\square$

Proposition 5.2

Given $\mathbf {H}$ and $\mathbf {G}^T$ from (44) and (33b) respectively, it holds that $\mathbf {H} = \mathbf {G}$.

Proof

From (44) and (46), it follows that

$$\begin{aligned} \mathbf {H}_e = - \bigl (\mathbf {A}_{L \hat{u}}^T \begin{bmatrix} \mathbf {B}_{\! e}^{-1} \end{bmatrix}_{12} + \mathbf {A}_{\hat{u}u} \begin{bmatrix} \mathbf {B}_{\! e}^{-1} \end{bmatrix}_{22} \bigr ) \mathbf {A}_{pu}^T \begin{bmatrix} \mathbf {S}_{\! e}^{-1} \end{bmatrix}_{12} + \mathbf {A}_{p\hat{u}}^T \begin{bmatrix} \mathbf {S}_{\! e}^{-1} \end{bmatrix}_{12} , \end{aligned}$$

(53)

for each element $\Omega _e$.

First, recall that the matrix $\mathbf {I} - \begin{bmatrix} \mathbf {S}_{\! e}\end{bmatrix}_{11}^{\dagger } \begin{bmatrix} \mathbf {S}_{\! e}\end{bmatrix}_{11}$ defines an orthogonal projector onto the kernel of $\begin{bmatrix} \mathbf {S}_{\! e}\end{bmatrix}_{11}$ (see Bernstein 2009, Sect. 6.1). As observed in the previous proposition, see (50), $\begin{bmatrix} \mathbf {S}_{\! e}\end{bmatrix}_{11}$ is the Schur complement of the velocity block of the matrix obtained from the discretization of an incompressible flow problem with purely Dirichlet boundary conditions. Thus, the kernel of $\begin{bmatrix} \mathbf {S}_{\! e}\end{bmatrix}_{11}$ contains all constant vectors representing the mean value of pressure. It follows that

$$\begin{aligned} \mathbf {a}_{\rho p} \bigl ( \mathbf {I} - \begin{bmatrix} \mathbf {S}_{\! e}\end{bmatrix}_{11}^{\dagger } \begin{bmatrix} \mathbf {S}_{\! e}\end{bmatrix}_{11} \bigr ) = \frac{1}{\texttt {n}_{\texttt {en}}} \mathbf {1}^T \end{aligned}$$

is the constant vector obtained as the average of 1 over the $\texttt {n}_{\texttt {en}}$ element nodes of $\Omega _e$ and, consequently, $\begin{bmatrix} \mathbf {S}_{\! e}^{-1} \end{bmatrix}_{12} = \mathbf {1}$. Moreover, since the kernel of the matrix $\mathbf {A}_{pu}^T$ also includes all constant vectors, $\mathbf {A}_{pu}^T \begin{bmatrix} \mathbf {S}_{\! e}^{-1} \end{bmatrix}_{12} = \mathbf {0}$. Hence, from (53), it follows that

$$\begin{aligned} \mathbf {H} = \begin{bmatrix} [\mathbf {H}_1]&[\mathbf {H}_2]&\ldots&[\mathbf {H}_{\texttt {n}_{\texttt {el}}}] \end{bmatrix} = \begin{bmatrix} [\mathbf {A}_{p\hat{u}}^T]_1 \mathbf {1}&[\mathbf {A}_{p\hat{u}}^T]_2 \mathbf {1}&\ldots&[\mathbf {A}_{p\hat{u}}^T]_{\texttt {n}_{\texttt {el}}} \mathbf {1} \end{bmatrix} \end{aligned}$$

which proves the statement. $\square$

When convection phenomena are neglected (Stokes flow), $\mathbf {A}_{\hat{u}u} = \mathbf {A}_{u \hat{u}}^T$ and the symmetry of $\widehat{\mathbf {K}}$ and the global matrix in (32) follows straightforwardly. For general incompressible flow problems, the matrix $\widehat{\mathbf {K}}$ is not symmetric but the off-diagonal blocks $\mathbf {G}$ and $\mathbf {G}^T$ are one the transpose of the other and the resulting global matrix maintains the above displayed saddle-point structure (Benzi et al. 2005).

Appendix: Implementation Details

In this Appendix, the matrices and vectors appearing in the discrete form of the HDG-Voigt approximation of the Oseen equations are detailed. The elemental variables $\varvec{u}$, p and $\varvec{L}$ are defined in a reference element $\widetilde{\Omega }(\varvec{\xi }), \ \varvec{\xi }= (\xi _1, \ldots , \xi _{\texttt {n}_{\texttt {sd}}})$ whereas the face variable $\varvec{\hat{u}}$, is defined on a reference face $\widetilde{\Gamma }(\varvec{\eta }), \ \varvec{\eta }= (\eta _1, \ldots , \eta _{\texttt {n}_{\texttt {sd}}-1})$ as

$$\begin{aligned} \varvec{u}(\varvec{\xi })&\simeq \sum _{j=1}^{\texttt {n}_{\texttt {en}}} \mathbf {u}_j N_j(\varvec{\xi }),&p(\varvec{\xi })&\simeq \sum _{j=1}^{\texttt {n}_{\texttt {en}}} {\mathrm {p}}_j N_j(\varvec{\xi }), \\ \varvec{L}(\varvec{\xi })&\simeq \sum _{j=1}^{\texttt {n}_{\texttt {en}}} \mathbf {L}_j N_j(\varvec{\xi }),&\varvec{\hat{u}}(\varvec{\eta })&\simeq \sum _{j=1}^{\texttt {n}_{\texttt {fn}}} \hat{\mathbf {u}}_j \hat{N}_j(\varvec{\eta }), \end{aligned}$$

where $\mathbf {u}_j, {\mathrm {p}}_j, \mathbf {L}_j$ and $\hat{\mathbf {u}}_j$ are the nodal values of the approximation, $\texttt {n}_{\texttt {en}}$ and $\texttt {n}_{\texttt {fn}}$ the number of nodes in the element and face, respectively and $N_j$ and $\hat{N}_j$ the polynomial shape functions in the reference element and face, respectively.

An isoparametric formulation is considered and the following transformation is used to map reference and local coordinates

$$\begin{aligned} \varvec{x}(\varvec{\xi }) = \sum _{k=1}^{\texttt {n}_{\texttt {en}}} \varvec{x}_k N_k(\varvec{\xi }), \end{aligned}$$

where the vector $\{\varvec{x}_k\}_{k=1,\ldots ,\texttt {n}_{\texttt {en}}}$ denotes the elemental nodal coordinates.

Following Sevilla et al. (2018), the matrices $\varvec{\nabla }_{\!\!\texttt {\small s}}$ and $\mathbf {N}$ in (18) and (22), respectively, are expressed in compact form as

$$\begin{aligned} \varvec{\nabla }_{\!\!\texttt {\small s}}= \sum _{k=1}^{\texttt {n}_{\texttt {sd}}} \mathbf {F}_k \frac{\partial }{\partial x_k}, \qquad \mathbf {N}= \sum _{k=1}^{\texttt {n}_{\texttt {sd}}} \mathbf {F}_k n_k , \end{aligned}$$

where the matrices $\mathbf {F}_k$ are defined as

$$\begin{aligned} \mathbf {F}_1 = \begin{bmatrix} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \end{bmatrix}^T , \quad \mathbf {F}_2 = \begin{bmatrix} 0 &{} 0 &{} 1 \\ 0 &{} 1 &{} 0 \end{bmatrix}^T \end{aligned}$$

in two dimensions and

$$\begin{aligned} \mathbf {F}_1 = \left[ \begin{array}{@{}c@{\;}c@{\;}c@{\;}c@{\;}c@{\;}c@{}} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 \end{array}\right] ^T , \; \mathbf {F}_2 = \left[ \begin{array}{@{}c@{\;}c@{\;}c@{\;}c@{\;}c@{\;}c@{}} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 \end{array}\right] ^T , \; \mathbf {F}_3 = \left[ \begin{array}{@{}c@{\;}c@{\;}c@{\;}c@{\;}c@{\;}c@{}} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \end{array}\right] ^T \end{aligned}$$

in three dimensions. Moreover, from the definition of $\mathbf {E}$ in (21), it holds $\mathbf {N}^T \mathbf {E}= \varvec{n}$ and $\varvec{\nabla }_{\!\!\texttt {\small s}}^T\mathbf {E}= \varvec{\nabla }$ for the gradient operator applied to a scalar function. The following compact forms of the shape functions and their derivatives are introduced

$$\begin{aligned} \varvec{\mathcal {N}}&:= \begin{bmatrix} N_1\mathbf {I}_{\texttt {n}_{\texttt {sd}}\!}&N_2\mathbf {I}_{\texttt {n}_{\texttt {sd}}\!}&\dots&N_{\texttt {n}_{\texttt {en}}}\mathbf {I}_{\texttt {n}_{\texttt {sd}}\!}\end{bmatrix}^T , \\ \varvec{\widehat{\mathcal {N}}}&:= \begin{bmatrix} \hat{N}_1\mathbf {I}_{\texttt {n}_{\texttt {sd}}\!}&\hat{N}_2\mathbf {I}_{\texttt {n}_{\texttt {sd}}\!}&\dots&\hat{N}_{\texttt {n}_{\texttt {fn}}}\mathbf {I}_{\texttt {n}_{\texttt {sd}}\!}\end{bmatrix}^T , \\ \varvec{\mathcal {N}}^{\tau }&:= \begin{bmatrix} N_1\tau \mathbf {I}_{\texttt {n}_{\texttt {sd}}\!}&N_2\tau \mathbf {I}_{\texttt {n}_{\texttt {sd}}\!}&\dots&N_{\texttt {n}_{\texttt {en}}}\tau \mathbf {I}_{\texttt {n}_{\texttt {sd}}\!}\end{bmatrix}^T , \\ \varvec{\widehat{\mathcal {N}}}^{\tau }&:= \begin{bmatrix} \hat{N}_1\tau \mathbf {I}_{\texttt {n}_{\texttt {sd}}\!}&\hat{N}_2 \tau \mathbf {I}_{\texttt {n}_{\texttt {sd}}\!}&\dots&\hat{N}_{\texttt {n}_{\texttt {fn}}} \tau \mathbf {I}_{\texttt {n}_{\texttt {sd}}\!}\end{bmatrix}^T , \\ \varvec{\mathcal {N}}^n&:= \begin{bmatrix} N_1\varvec{n}&N_2\varvec{n}&\dots&N_{\texttt {n}_{\texttt {en}}}\varvec{n}\end{bmatrix}^T , \\ \varvec{\mathcal {M}}&:= \begin{bmatrix} N_1\mathbf {I}_{\texttt {m}_{\texttt {sd}}}&N_2\mathbf {I}_{\texttt {m}_{\texttt {sd}}}&\dots&N_{\texttt {n}_{\texttt {en}}}\mathbf {I}_{\texttt {m}_{\texttt {sd}}}\end{bmatrix}^T , \\ \varvec{\mathcal {N}}^a&:= \begin{bmatrix} N_1(\tau - \varvec{\hat{a}}{\,\cdot \,} \varvec{n}) \mathbf {I}_{\texttt {n}_{\texttt {sd}}\!}&N_2 (\tau - \varvec{\hat{a}}{\,\cdot \,} \varvec{n}) \mathbf {I}_{\texttt {n}_{\texttt {sd}}\!}&\dots&N_{\texttt {n}_{\texttt {en}}} (\tau - \varvec{\hat{a}}{\,\cdot \,} \varvec{n}) \mathbf {I}_{\texttt {n}_{\texttt {sd}}\!}\end{bmatrix}^T , \\ \varvec{\widehat{\mathcal {N}}}^a&:= \begin{bmatrix} \hat{N}_1 (\tau - \varvec{\hat{a}}{\,\cdot \,} \varvec{n}) \mathbf {I}_{\texttt {n}_{\texttt {sd}}\!}&\hat{N}_2 (\tau - \varvec{\hat{a}}{\,\cdot \,} \varvec{n}) \mathbf {I}_{\texttt {n}_{\texttt {sd}}\!}&\dots&\hat{N}_{\texttt {n}_{\texttt {fn}}} (\tau - \varvec{\hat{a}}{\,\cdot \,} \varvec{n}) \mathbf {I}_{\texttt {n}_{\texttt {sd}}\!}\end{bmatrix}^T , \\ \varvec{\mathcal {Q}}&:= \begin{bmatrix} (\mathbf {J}^{-1} \varvec{\nabla }\! N_1)^T&(\mathbf {J}^{-1} \varvec{\nabla }\! N_2)^T&\dots&(\mathbf {J}^{-1} \varvec{\nabla }\! N_{\texttt {n}_{\texttt {en}}})^T\end{bmatrix}^T , \\ \varvec{\mathcal {Q}}^a&:= \begin{bmatrix} \varvec{a} \,\cdot \, (\mathbf {J}^{-1} \varvec{\nabla }\! N_1)&\varvec{a} \,\cdot \, (\mathbf {J}^{-1} \varvec{\nabla }\! N_2)&\dots&\varvec{a} \,\cdot \, (\mathbf {J}^{-1} \varvec{\nabla }\! N_{\texttt {n}_{\texttt {en}}}) \end{bmatrix}^T , \end{aligned}$$

where $\varvec{n}$ is the outward unit normal vector to a face, $\varvec{a}$ is the convection field evaluated in the reference element, and $\varvec{\hat{a}}$ is the convection field evaluated on the reference face. Moreover, for each spatial dimension, that is, for $k=1,\ldots ,\texttt {n}_{\texttt {sd}}$, define

$$\begin{aligned} \varvec{\mathcal {N}}^{\mathrm {D}}_k&\,{:=}\, \begin{bmatrix} N_1 n_k \mathbf {F}^T_k \mathbf {D}^{1/2}&N_2 n_k \mathbf {F}^T_k \mathbf {D}^{1/2}&\dots&N_{\texttt {n}_{\texttt {fn}}} n_k \mathbf {F}^T_k \mathbf {D}^{1/2}\end{bmatrix}^T, \\ \varvec{\widehat{\mathcal {N}}}^{\mathrm {D}}_k&\,{:=}\, \begin{bmatrix} \hat{N}_1 n_k \mathbf {F}^T_k \mathbf {D}^{1/2}&\hat{N}_2 n_k \mathbf {F}^T_k \mathbf {D}^{1/2}&\dots&\hat{N}_{\texttt {n}_{\texttt {fn}}} n_k \mathbf {F}^T_k \mathbf {D}^{1/2}\end{bmatrix}^T, \\ \varvec{\mathcal {Q}}^{\mathrm {D}}_k&\,{:=}\, \left[ \begin{array}{@{}c@{\;}c@{\;}c@{\;}c@{}} [ \mathbf {J}^{-1} \varvec{\nabla }\! N_1 ]_{k} \mathbf {F}^T_k \mathbf {D}^{1/2}&[ \mathbf {J}^{-1} \varvec{\nabla }\! N_2 ]_{k} \mathbf {F}^T_k \mathbf {D}^{1/2}&\dots&[ \mathbf {J}^{-1} \varvec{\nabla }\! N_{\texttt {n}_{\texttt {en}}} ]_{k} \mathbf {F}^T_k \mathbf {D}^{1/2}\end{array}\right] ^T , \end{aligned}$$

where $n_k$ is the k-th components of the outward unit normal vector $\varvec{n}$ to a face and $\mathbf {J}$ is the Jacobian of the isoparametric transformation.

The discretization of (28a) leads to the following matrices and vector

$$\begin{aligned}{}[\mathbf {A}_{LL} ]_e&= -\sum _{\texttt {g}=1}^{\texttt {n}_{\texttt {ip}}^{\texttt {e}}}\varvec{\mathcal {M}}(\varvec{\xi }^\texttt {e}_\texttt {g}) \varvec{\mathcal {M}}^T\!(\varvec{\xi }^\texttt {e}_\texttt {g}) |\mathbf {J}(\varvec{\xi }^\texttt {e}_\texttt {g})| w^\texttt {e}_\texttt {g}, \\ [\mathbf {A}_{Lu}]_e&= \sum _{k=1}^{\texttt {n}_{\texttt {sd}}} \sum _{\texttt {g}=1}^{\texttt {n}_{\texttt {ip}}^{\texttt {e}}}\varvec{\mathcal {Q}}^{\mathrm {D}}_k(\varvec{\xi }^\texttt {e}_\texttt {g}) \varvec{\mathcal {N}}^T\!(\varvec{\xi }^\texttt {e}_\texttt {g}) |\mathbf {J}(\varvec{\xi }^\texttt {e}_\texttt {g})| w^\texttt {e}_\texttt {g}, \\ [\mathbf {A}_{L \hat{u}}]_e&= \sum _{f=1}^{\texttt {n}_{\texttt {fa}}^e} \biggl ( \sum _{k=1}^{\texttt {n}_{\texttt {sd}}} \sum _{\texttt {g}=1}^{\texttt {n}_{\texttt {ip}}^{\texttt {f}}}\varvec{\mathcal {N}}^{\mathrm {D}}_k (\varvec{\xi }^\texttt {f}_\texttt {g}) \varvec{\widehat{\mathcal {N}}}^T\!(\varvec{\xi }^\texttt {f}_\texttt {g}) |\mathbf {J}(\varvec{\xi }^\texttt {f}_\texttt {g})| w^\texttt {f}_\texttt {g}\biggr ) \bigl (1 - \chi _{\Gamma _D}(f) \bigr ), \\ [\mathbf {f}_{L}]_e&= \sum _{f=1}^{\texttt {n}_{\texttt {fa}}^e} \biggl ( \sum _{k=1}^{\texttt {n}_{\texttt {sd}}} \sum _{\texttt {g}=1}^{\texttt {n}_{\texttt {ip}}^{\texttt {f}}}\varvec{\mathcal {N}}^{\mathrm {D}}_k(\varvec{\xi }^\texttt {f}_\texttt {g}) \varvec{u}_D \!\bigl (\varvec{x}(\varvec{\xi }^\texttt {f}_\texttt {g}) \bigr ) |\mathbf {J}(\varvec{\xi }^\texttt {f}_\texttt {g})| w^\texttt {f}_\texttt {g}\biggr ) \chi _{\Gamma _D}(f) , \end{aligned}$$

where $\texttt {n}_{\texttt {fa}}^e$ is the number of faces $\Gamma _{e,f}, \ f = 1,\dots ,\texttt {n}_{\texttt {fa}}^e$, of the element $\Omega _e$, $\varvec{\xi }^\texttt {e}_\texttt {g}$ and $w^\texttt {e}_\texttt {g}$ (resp., $\varvec{\xi }^\texttt {f}_\texttt {g}$ and $w^\texttt {f}_\texttt {g}$ ) are the $\texttt {n}_{\texttt {ip}}^{\texttt {e}}$ (resp., $\texttt {n}_{\texttt {ip}}^{\texttt {f}}$) integration points and weights defined on the reference element (resp., face) and $\chi _{\Gamma _D}$ is the indicator function of the boundary $\Gamma _D$, namely

$$\begin{aligned} \chi _{\Gamma _D}(f) = {\left\{ \begin{array}{ll} 1 &{} \text { if } \Gamma _{e,f} \cap \Gamma _D \ne \emptyset \\ 0 &{} \text { otherwise} \end{array}\right. } \end{aligned}$$

Similarly, from the discretization of (28b) the following matrices and vectors are obtained

$$\begin{aligned}{}[\mathbf {A}_{uu}]_e&= - \sum _{\texttt {g}=1}^{\texttt {n}_{\texttt {ip}}^{\texttt {e}}}\varvec{\mathcal {Q}}^a(\varvec{\xi }^\texttt {e}_\texttt {g}) \varvec{\mathcal {N}}^T\!(\varvec{\xi }^\texttt {e}_\texttt {g}) |\mathbf {J}(\varvec{\xi }^\texttt {e}_\texttt {g})| w^\texttt {e}_\texttt {g}\\&\quad + \sum _{f=1}^{\texttt {n}_{\texttt {fa}}^e} \sum _{\texttt {g}=1}^{\texttt {n}_{\texttt {ip}}^{\texttt {f}}}\varvec{\mathcal {N}}^{\tau }\!(\varvec{\xi }^\texttt {f}_\texttt {g}) \varvec{\mathcal {N}}^T\!(\varvec{\xi }^\texttt {f}_\texttt {g}) |\mathbf {J}(\varvec{\xi }^\texttt {f}_\texttt {g})| w^\texttt {f}_\texttt {g}, \\ [\mathbf {A}_{u \hat{u}}]_e&= \sum _{f=1}^{\texttt {n}_{\texttt {fa}}^e} \biggl ( \sum _{\texttt {g}=1}^{\texttt {n}_{\texttt {ip}}^{\texttt {f}}}\varvec{\mathcal {N}}^a\!(\varvec{\xi }^\texttt {f}_\texttt {g}) \varvec{\widehat{\mathcal {N}}}^T\!(\varvec{\xi }^\texttt {f}_\texttt {g}) |\mathbf {J}(\varvec{\xi }^\texttt {f}_\texttt {g})| w^\texttt {f}_\texttt {g}\biggr ) \bigl (1 - \chi _{\Gamma _D}(f) \bigr ), \\ [\mathbf {f}_{u}]_e&= \sum _{\texttt {g}=1}^{\texttt {n}_{\texttt {ip}}^{\texttt {e}}}\varvec{\mathcal {N}}(\varvec{\xi }^\texttt {e}_\texttt {g})\varvec{s}\!\bigl (\varvec{x}(\varvec{\xi }^\texttt {e}_\texttt {g}) \bigr ) |\mathbf {J}(\varvec{\xi }^\texttt {e}_\texttt {g})| w^\texttt {e}_\texttt {g}\\&\quad + \sum _{f=1}^{\texttt {n}_{\texttt {fa}}^e} \biggl ( \sum _{\texttt {g}=1}^{\texttt {n}_{\texttt {ip}}^{\texttt {f}}}\varvec{\mathcal {N}}^a\!(\varvec{\xi }^\texttt {f}_\texttt {g}) \varvec{u}_D\!\bigl (\varvec{x}(\varvec{\xi }^\texttt {f}_\texttt {g}) \bigr ) |\mathbf {J}(\varvec{\xi }^\texttt {f}_\texttt {g})| w^\texttt {f}_\texttt {g}\biggr ) \chi _{\Gamma _D}(f) . \end{aligned}$$

The discrete forms of the incompressibility constraint in (28c) and the restriction in (28d) feature the following matrices and vector

$$\begin{aligned}{}[\mathbf {A}_{pu}]_e&= \sum _{\texttt {g}=1}^{\texttt {n}_{\texttt {ip}}^{\texttt {e}}}\varvec{\mathcal {N}}(\varvec{\xi }^\texttt {e}_\texttt {g}) \varvec{\mathcal {Q}}^T\!(\varvec{\xi }^\texttt {e}_\texttt {g}) |\mathbf {J}(\varvec{\xi }^\texttt {e}_\texttt {g})| w^\texttt {e}_\texttt {g}, \\ [\mathbf {A}_{p \hat{u}}]_e&= \sum _{f=1}^{\texttt {n}_{\texttt {fa}}^e} \biggl ( \sum _{\texttt {g}=1}^{\texttt {n}_{\texttt {ip}}^{\texttt {f}}}\varvec{\mathcal {N}}^n(\varvec{\xi }^\texttt {f}_\texttt {g}) \varvec{\widehat{\mathcal {N}}}^T\!(\varvec{\xi }^\texttt {f}_\texttt {g}) |\mathbf {J}(\varvec{\xi }^\texttt {f}_\texttt {g})| w^\texttt {f}_\texttt {g}\biggr ) \bigl (1 - \chi _{\Gamma _D}(f) \bigr ) , \\ [\mathbf {f}_{p}]_e&= \sum _{f=1}^{\texttt {n}_{\texttt {fa}}^e} \biggl ( \sum _{\texttt {g}=1}^{\texttt {n}_{\texttt {ip}}^{\texttt {f}}}\varvec{\mathcal {N}}^n(\varvec{\xi }^\texttt {f}_\texttt {g}) \varvec{u}_D\!\bigl (\varvec{x}(\varvec{\xi }^\texttt {f}_\texttt {g}) \bigr ) |\mathbf {J}(\varvec{\xi }^\texttt {f}_\texttt {g})| w^\texttt {f}_\texttt {g}\biggr ) \chi _{\Gamma _D}(f) , \\ [\mathbf {a}_{\rho p}]_e&= \sum _{f=1}^{\texttt {n}_{\texttt {fa}}^e} \sum _{\texttt {g}=1}^{\texttt {n}_{\texttt {ip}}^{\texttt {f}}}\varvec{\mathcal {N}}(\varvec{\xi }^\texttt {f}_\texttt {g}) \mathbf {1} |\mathbf {J}(\varvec{\xi }^\texttt {f}_\texttt {g})| w^\texttt {f}_\texttt {g}, \end{aligned}$$

Finally, the matrices and vectors resulting from the discretization of the global problem in (29a) are

$$\begin{aligned}{}[\mathbf {A}_{\hat{u} \hat{u}}]_e&= - \sum _{f=1}^{\texttt {n}_{\texttt {fa}}^e} \biggl ( \sum _{\texttt {g}=1}^{\texttt {n}_{\texttt {ip}}^{\texttt {f}}}\varvec{\widehat{\mathcal {N}}}^{\tau }\!(\varvec{\xi }^\texttt {f}_\texttt {g}) \varvec{\widehat{\mathcal {N}}}^T(\varvec{\xi }^\texttt {f}_\texttt {g}) |\mathbf {J}(\varvec{\xi }^\texttt {f}_\texttt {g})| w^\texttt {f}_\texttt {g}\biggr ) \chi _{\Gamma }(f) \\&\quad - \sum _{f=1}^{\texttt {n}_{\texttt {fa}}^e} \biggl ( \sum _{\texttt {g}=1}^{\texttt {n}_{\texttt {ip}}^{\texttt {f}}}\varvec{\widehat{\mathcal {N}}}^a\!(\varvec{\xi }^\texttt {f}_\texttt {g}) \varvec{\widehat{\mathcal {N}}}^T(\varvec{\xi }^\texttt {f}_\texttt {g}) |\mathbf {J}(\varvec{\xi }^\texttt {f}_\texttt {g})| w^\texttt {f}_\texttt {g}\biggr ) \chi _{\Gamma _N}(f) , \\ [\mathbf {A}_{\hat{u} u}]_e&= \sum _{f=1}^{\texttt {n}_{\texttt {fa}}^e} \biggl ( \sum _{\texttt {g}=1}^{\texttt {n}_{\texttt {ip}}^{\texttt {f}}}\varvec{\widehat{\mathcal {N}}}^{\tau }\!(\varvec{\xi }^\texttt {f}_\texttt {g}) \varvec{\mathcal {N}}^T\!(\varvec{\xi }^\texttt {f}_\texttt {g}) |\mathbf {J}(\varvec{\xi }^\texttt {f}_\texttt {g})| w^\texttt {f}_\texttt {g}\biggr ) \bigl (1 - \chi _{\Gamma _D}(f) \bigr ) , \\ [\mathbf {f}_{\hat{u}}]_e&= - \sum _{f=1}^{\texttt {n}_{\texttt {fa}}^e} \biggl ( \sum _{\texttt {g}=1}^{\texttt {n}_{\texttt {ip}}^{\texttt {f}}}\varvec{\widehat{\mathcal {N}}}(\varvec{\xi }^\texttt {f}_\texttt {g}) \varvec{t}\!\bigl (\varvec{x}(\varvec{\xi }^\texttt {f}_\texttt {g}) \bigr ) |\mathbf {J}(\varvec{\xi }^\texttt {f}_\texttt {g})| w^\texttt {f}_\texttt {g}\biggr ) \chi _{\Gamma _N}(f) , \end{aligned}$$

where $\chi _{\Gamma }$ and $\chi _{\Gamma _N}$ are the indicator functions of the internal skeleton $\Gamma$ and the Neumann boundary $\Gamma _N$, respectively.