Lower Bounds for Eigenvalues of Elliptic Operators: By Nonconforming Finite Element Methods

Abstract

The paper is to introduce a new systematic method that can produce lower bounds for eigenvalues. The main idea is to use nonconforming finite element methods. The conclusion is that if local approximation properties of nonconforming finite element spaces are better than total errors (sums of global approximation errors and consistency errors) of nonconforming finite element methods, corresponding methods will produce lower bounds for eigenvalues. More precisely, under three conditions on continuity and approximation properties of nonconforming finite element spaces we analyze abstract error estimates of approximate eigenvalues and eigenfunctions. Subsequently, we propose one more condition and prove that it is sufficient to guarantee nonconforming finite element methods to produce lower bounds for eigenvalues of symmetric elliptic operators. We show that this condition hold for most low-order nonconforming finite elements in literature. In addition, this condition provides a guidance to modify known nonconforming elements in literature and to propose new nonconforming elements. In fact, we enrich locally the Crouzeix-Raviart element such that the new element satisfies the condition; we also propose a new nonconforming element for second order elliptic operators and prove that it will yield lower bounds for eigenvalues. Finally, we prove the saturation condition for most nonconforming elements.

Appendix A: The Comment for the Saturation Condition of the Singular Case

We need the concept of the interpolation space. Let $X$, $Y$ be a pair of normed linear spaces. We shall assume that $Y$ is continuously embedded in $X$ with $Y\subset X$ and $\Vert \cdot \Vert _X\lesssim \Vert \cdot \Vert _Y$. For any $t\ge 0$, we define the $K-$functional

$$\begin{aligned} K(f,t)=K(f,t,X,Y)=\inf \limits _{g\in Y}\Vert f-g\Vert _X+t|g|_Y, \end{aligned}$$

(10.1)

where $\Vert \cdot \Vert _X$ is the norm on $X$ and $|\cdot |_Y$ is a semi-norm on $Y$. The function $K(f,.)$ is defined on ${\mathbb {R}}_+$ and is monotone and concave (being the pointwise infimum of linear functions). If $0<\theta <1$ and $1<q\le \infty$, the interpolation space $(X,Y)_{\theta , q}$ is defined as the set of all functions $f\in X$ such that [6, 15, 16]

$$\begin{aligned} |f|_{(X,Y)_{\theta , q}}=\left\{ \begin{array}{l} (\sum \limits _{k=0}^{\infty }[2^{(s+\epsilon )k\theta }K(f,2^{-k(s+\epsilon )})]^q)^{1/q},\quad 0<q<\infty ,\\ \sup \limits _{k\ge 0}2^{k(s+\epsilon )\theta }K(f, 2^{-k(s+\epsilon )}), \quad q=\infty , \end{array}\right. \end{aligned}$$

(10.2)

is finite for some $0<s+\epsilon \le 1$.

1.1 An Abstract Theory

We assume that $u\in H^{m+s}(\Omega )$ with $0<s<1$ and $V_h$ is some nonconforming or conforming approximation space to the space $H^m(\Omega )$.

Then the following conditions imply in some sense the saturation condition for the singular case:

• (H6) There exists a piecewise polynomial space $V_{m+1,h}^c\subset H^{m+1}(\Omega )$ such that $V_{m+1,h}^c\subset V_{m+1,h/2}^c$ when ${\mathcal {T}}_{h/2}$ is some nested conforming refinement of ${\mathcal {T}}_h$;

• (H7) There holds the following Berstein inequality

  $$\begin{aligned} |v|_{H^{m+s+\epsilon }(\Omega )}\lesssim h^{-(s+\epsilon )}|v|_{H^m(\Omega )}\text { for any }v\in V_{m+1,h}^c; \end{aligned}$$

  (10.3)

• (H8) There exists some quasi-interpolation operator $\Pi ^c: V_h\rightarrow V_{m+1, h}^c$ such that

  $$\begin{aligned} \Vert D^m(u-\Pi ^c u_h)\Vert _{L^2(\Omega )}\lesssim h^{s+\epsilon } \end{aligned}$$

  (10.4)

  provided that $\Vert D_h^m(u-u_h)\Vert _{L^2(\Omega )}\lesssim h^{s+\epsilon }$ with $\epsilon >0$ and $s+\epsilon \le 1$.

Theorem 10.1

Suppose the eigenfunction $u\in H^{m+s}(\Omega )$ with $0<s<1$. Under conditions (H6)–(H8), there exist meshes such that the following saturation condition holds

$$\begin{aligned} h^s\lesssim \Vert D_h^m(u-u_h)\Vert _{L^2(\Omega )}. \end{aligned}$$

(10.5)

Proof

We assume that the saturation condition $h^{s}\lesssim \Vert D_h^m(u-u_h)\Vert _{L^2(\Omega )}$ does not hold for any mesh ${\mathcal {T}}_h$ with the meshsize $h$. In other word, we have

$$\begin{aligned} \Vert D_h^m(u-u_h)\Vert _{L^2(\Omega )}\lesssim h^{s+\epsilon }, \end{aligned}$$

(10.6)

for some $\epsilon >0$. In the following, we assume that $s+\epsilon \le 1$. By the condition (H8), we have

$$\begin{aligned} \inf \limits _{v\in V_{m+1,h}^c}\Vert D^m(u-v)\Vert _{L^2(\Omega )}\lesssim \Vert D_h^m(u-\Pi ^c u_h)\Vert _{L^2(\Omega )}\lesssim h^{s+\epsilon }. \end{aligned}$$

(10.7)

Take $X=H^m(\Omega )$ and $Y=H^{m+s+\epsilon }(\Omega )$. The inequality (10.7) is the usual Jackson inequality and the inequality (10.3) is the Berstein inequality in the context of the approximation theory [15, 16]. We can follow the idea of [16, Theorem 5.1, Chapter 7] to estimate terms like $K(u, 2^{-{\ell }(s+\epsilon )})$ for any positive integer $\ell$. In fact, let $\varphi _k\in V_{m+1, 2^{-k}}$ be the best approximation to $u$ in the sense that $\Vert D^m(u-\varphi _k)\Vert _{L^2(\Omega )}=\inf \limits _{v\in V_{m+1,2^{-k}}}\Vert D^m(u-v)\Vert _{L^2(\Omega )}$, $k\ge 1$. Let $\psi _k=\varphi _k-\varphi _{k-1}$, $k=1, 2,\ldots ,$ where $\psi _0=\varphi _0=0$. Then we have

$$\begin{aligned} \Vert D^m\psi _k\Vert _{L^2(\Omega )}\le \Vert D^m(u-\varphi _k)\Vert _{L^2(\Omega )}+\Vert D^m(u-\varphi _{k-1})\Vert _{L^2(\Omega )}\lesssim 2^{-k(s+\epsilon )}. \end{aligned}$$

(10.8)

Since $\varphi _{\ell }=\sum \limits _{k=0}^{\ell }\psi _k$ and $|\psi _0|_{H^{m+s+\epsilon }(\Omega )}=0$, it follows from (10.3), (10.7) and (10.8) that

$$\begin{aligned} K(u, 2^{-(s+\epsilon )\ell })&\le \Vert u-\varphi _{\ell }\Vert _{H^{m}(\Omega )}+2^{-(s+\epsilon )\ell }| \varphi _{\ell }|_{H^{m+s+\epsilon }}\nonumber \\&\lesssim 2^{-(s+\epsilon )\ell }+2^{-(s+\epsilon )\ell }\sum \limits _{k=1}^{\ell } 2^{k(s+\epsilon )}|\psi _k|_{H^m(\Omega )}\nonumber \\&\lesssim \ell 2^{-(s+\epsilon )\ell }. \end{aligned}$$

(10.9)

$$\begin{aligned} |u|_{(H^{m}(\Omega ), H^{m+s+\epsilon }(\Omega ))_{\theta , 2}}&= \bigg (\sum \limits _{k=0}^{\infty }\big [2^{k(s+\epsilon )\theta }K(u,2^{-k(s+\epsilon )})\big ]^2\bigg )^{1/2}\nonumber MYAMP]\lesssim&\bigg (\sum \limits _{k=0}^{\infty }\big [k2^{k(s+\epsilon )(\theta -1)}\big ]^2\bigg )^{1/2}. \end{aligned}$$

(10.10)

Let $\theta =1-\epsilon _0$ with $\epsilon _0>0$ such that $\epsilon -(s+\epsilon )\epsilon _0>0$. This leads to

$$\begin{aligned} |u|_{(H^{m}(\Omega ), H^{m+s+\epsilon }(\Omega ))_{\theta , 2}} \lesssim \bigg (\sum \limits _{k=0}^{\infty }\big [k2^{-k(s+\epsilon )\epsilon _0}\big ] ^2\bigg )^{1/2}<\infty . \end{aligned}$$

(10.11)

This proves that $u\in H^{m+(1-\epsilon _0)(s+\epsilon )}(\Omega )$ which is a proper subspace of $H^{m+s}(\Omega )$ since $\epsilon -(s+\epsilon )\epsilon _0>0$, which contradicts with the fact that we only have the regularity $u\in H^{m+s}(\Omega )$. $\square$

1.2 Proofs for the Conditions (H6)–(H8)

It follows from Davydov [14] that there exist piecewise polynomial spaces $V_{m+1,h}^c$ with nodal basis over ${\mathcal {T}}_h$ such that $V_{m+1,h}^c$ are nested and conforming in the sense that $V_{m+1,h}^c\subset V_{m+1,h/2}^c\subset H^m(\Omega )$ for any $1\le n$ and $m\le n$.

This result actually proves the conditions (H6) and (H7). The proof of (H8) needs the interpolation of $V_h$ into the conforming finite element space. To this end, we introduce the projection average interpolation operator of Brenner [8], Shi and Wang [36].

Let $V_{m+1,h}^c$ be a conforming finite element space defined by $(K,P_K^c,D^c_K)$, where $D_K^c$ is the vector functional and the components of $D^c_K$ are defined as follows: for any $v\in C^{\kappa }(K)$

$$\begin{aligned} d_{i,K}(v){:=}\left\{ \begin{array}{l@{\quad }l} D_{i,K}v(a_{i,K}) &{} 1\le i\le k_1,\\ \frac{1}{|F_{i,K}|}\int \limits _{F_{i,K}}D_{i,K}v df &{} k_1< i\le k_2,\\ \frac{1}{|K|}\int \limits _KD_{i,K}v dx &{} k_2< i\le L,\\ \end{array}\right. \end{aligned}$$

(*)

where $a_{i,K}$ are points in $K$, $F_{i,K}$ are non zero-dimensional faces of $K$. $\kappa {:=}\max \limits _{1\le i\le L}k(i)$ where $k(i)$ orders of derivatives used in degrees of freedom $D_{i,K}{:=}\sum \limits _{|\alpha |=k(i)}\eta _{i,\alpha , K}\partial ^{\alpha },1\le i\le L$, $\eta _{i,\alpha , K}$ are constants which depend on $i, \alpha , \text { and } K$.

Let $\omega (a)$ denote the union of elements that share point $a$ and $\omega (F)$ the union of elements having in common the face $F$. Let $N(a)$ and $N(F)$ denote the number of elements in $\omega (a)$ and $\omega (F)$, respectively. For any $v\in V_h$, define the projection average interpolation operator $\Pi ^c: V_h\rightarrow V_{m+1,h}^c$ by

• (1) for $1\le i\le k_1$, if $a_{i,K}\in \partial \Omega$ and $d_{i,K}(\phi )=0$ for any $\phi \in C^{\kappa }(\overline{\Omega })\cap V$, then $d_{i,K}(\Pi ^c v|_K){:=}0$; otherwise

  $$\begin{aligned} d_{i,K}(\Pi ^c v|_K){:=}\frac{1}{N(a_{i,K})}\sum _{K'\in \omega (a_{i,K})}D_{i,K}(v|_{K'})(a_{i,K}); \end{aligned}$$

• (2) for $k_1< i\le k_2$, if $F_{i,K}\subset \partial \Omega$ and $d_{i,K}(\phi )=0$ for any $\phi \in C^{\kappa }(\overline{\Omega })\cap V$, then $d_{i,K}(\Pi ^c v|_K){:=}0$; otherwise

  $$\begin{aligned} d_{i,K}(\Pi ^c v|_K){:=}\frac{1}{N(F_{i,K})}\sum _{K'\in \omega (F_{i,K})}\frac{1}{|F_{i,K}|}\int \limits _{F_{i,K}}D_{i,K}(v|_{K'})(a_{i,K})df; \end{aligned}$$

• (3) for $k_2< i\le L$

  $$\begin{aligned} d_{i, K}(\Pi ^c v|_K){:=}\frac{1}{|K|}\int \limits _{K}D_{i,K}(v|_{K})dx. \end{aligned}$$

Lemma 10.2

For all nonconforming element spaces under consideration, there exists $r\in \mathbb {N}, r\geqslant m$ such that $V_h|_K\subset P_r(K)\subset P^c_K$. Then, for $m<k\leqslant min\{r+1,2m\}$, $0\leqslant l\leqslant m$, $\alpha =(\alpha _1,\ldots ,\alpha _n)$, it holds that

$$\begin{aligned} \Vert D_h^m(v_h-\Pi ^c v_h)\Vert ^2_{L^2(\Omega )}&\lesssim \sum _{K\in \mathcal {T}_h}\left( \sum ^{k-1}_{j=m}h_K^{2(j-m)+1}\sum _{F\subset \partial K/\partial \Omega }\sum _{|\alpha |=j}\Vert [\partial ^{\alpha }v_h]\Vert ^2_{0,F}\right. \\&\left. +\,h_K\sum _{F\subset \partial K\cap \partial \Omega }\sum _{|\alpha |=m,\alpha _1<m}\Vert \frac{\partial ^{|\alpha |}v_h}{\partial \nu _F^{\alpha _1}\partial \tau _{F,2}^{\alpha _2}\cdots \partial \tau _{F,n}^{\alpha _n}} \Vert ^2_{0,F}\right) , \end{aligned}$$

where $\tau _{F,2},\ldots ,\tau _{F,n}$ are $n-1$ orthonormal tangent vectors of $F$.

Proof

Since $V_h|_K\subset P_r(K)\subset P^c_K$, a slight modification of the argument in [36, Lemma 5.6.4] can prove the desired result; see also Brenner [8] for the proof of the nonconforming linear element with $m=1$. $\square$

The remaining proof is based on bubble function techniques, see Carstensen and Hu [11] for a posteriori error analysis of second order problems, see Gudi [19], Hu and Shi [21] for a posteriori error analysis of fourth order problems. Let $v_h=u_h$ in the above lemma. Such an analysis leads to

$$\begin{aligned} \Vert D_h^m(\Pi ^c u_h-u_h)\Vert _{L^2(\Omega )}\lesssim \Vert D_h^m(u-u_h)\Vert _{L^2(\Omega )}\lesssim h^{s+\epsilon }. \end{aligned}$$

(10.12)