Convergence rate of the finite element approximation for extremizers of Sobolev inequalities on 2D convex domains

Abstract

We investigate a FEM-based numerical scheme approximating extremal functions of the Sobolev inequalities. The main result of this paper shows that if the domain is polygonal and convex in $\mathbb {R}^2$, then the convergence rate of a finite element solution to an exact extremal function is $O(h^2)$ in the $L^2$ norm, and it is O(h) in the $H^1$ norm, where h denotes the mesh size of a triangulation of the domain.

Appendix A: Analytic tools

In the appendix, we arrange the necessary auxiliary tools to handle the analytic issues that arise when we prove our main results.

Proposition A.1

([2, 3]) For any $u \in H_0^1(\Omega )$, we define $P_h(u)$ by the $H_0^1 (\Omega )$-projection of u onto $V_h$. In other words, $P_h(u)$ is a unique element in $V_h$ that satisfies

$$\begin{aligned} \int _{\Omega }\nabla u\cdot \nabla \phi _h\,dx = \int _{\Omega } \nabla P_h(u)\cdot \nabla \phi _h\,dx \ \text { for all } \phi _h \in V_h. \end{aligned}$$

Then the following estimates hold:

$$\begin{aligned} \Vert u-P_h(u)\Vert _{H_0^1(\Omega )}= & {} o(1), \quad \text {and} \\ \Vert u-P_h(u)\Vert _{L^2(\Omega )}= & {} O (h) \Vert u\Vert _{H^1_0(\Omega )} \quad \text {as } h \rightarrow 0. \end{aligned}$$

If $u \in H_0^1(\Omega ) \cap H^2(\Omega )$ the following estimates hold:

$$\begin{aligned} \Vert u-P_h(u)\Vert _{H^1(\Omega )} = O(h) \Vert u\Vert _{H^2(\Omega )} \quad \text {and} \quad \Vert u-P_h(u)\Vert _{L^2(\Omega )} = O(h^2) \Vert u\Vert _{H^2(\Omega )} \quad \text {as } h \rightarrow 0. \end{aligned}$$

If $u \in W^{1,q}_0(\Omega )$ for some $q \ge 2$, the following estimate holds:

$$\begin{aligned} \Vert P_h(u)\Vert _{W^{1,q}(\Omega )} \le C\Vert u\Vert _{W^{1,q}(\Omega )} \end{aligned}$$

for some $C > 0$ independent of h (refer to [Theorem 8.5.3] in [3]).

Proposition A.2

([9]) Let $\Omega \subset \mathbb {R}^2$ be a bounded convex domain with a polygonal boundary. For a given $f \in L^2(\Omega )$, let $u \in H^1_0(\Omega )$ be a weak solution of the problem

$$\begin{aligned} -\Delta u = f \ \text { in } \Omega , \quad u \in H^1_0(\Omega ) \end{aligned}$$

Then u belongs to $H^2(\Omega )$, and there exists a constant $C>0$ such that

$$\begin{aligned} \Vert u\Vert _{H^2 (\Omega )} \le C\Vert f\Vert _{L^2 (\Omega )}. \end{aligned}$$

Proposition A.3

([6, 11]) Let $\Omega \subset \mathbb {R}^2$ be a bounded convex domain and $p \in (2, \infty )$. Let U be a minimizer of the problem

$$\begin{aligned} C(\Omega , p) = \inf \left\{ \frac{\Vert \nabla u\Vert _{L^2(\Omega )}}{\Vert u\Vert _{L^p(\Omega )}} ~\Big |~ u \in H^1_0(\Omega ), u \ne 0\right\} \end{aligned}$$

satisfying

$$\begin{aligned} -\Delta u = |u|^{p-2}u \ \text { in } \Omega . \end{aligned}$$

(A.1)

Then, the following holds:

• (i) U is sign definite and unique up to a sign.

• (ii) U is non-degenerate. In other words, the linearized equation of (A.1) at U, i.e.,

  $$\begin{aligned} \Delta \phi +(p-1)U^{p-2}\phi = 0 \ \text { in } \Omega , \quad \phi \in H^1_0(\Omega ) \end{aligned}$$

  admits only the trivial solution.

• (iii) The following inequality

  $$\begin{aligned} \int _{\Omega } |\nabla \phi |^2 - (p-1) U^{p-2} \phi ^2 dx \ge C \int _{\Omega } |\nabla \phi |^2 dx \end{aligned}$$

  (A.2)

  holds true for any $\phi \in H^1_0(\Omega )$ that satisfies $\langle \phi , U\rangle _{H^1_0(\Omega )} = 0$ and some $C > 0$ independent of $\phi$.

Remark A.4

The statements (i) and (ii) are proved in [11]. The statement (iii) is a natural consequence of (ii). We refer to [6] for the rigorous arguments of the proof.