Maximum-norm stability of the finite element method for the Neumann problem in nonconvex polygons with locally refined mesh

Abstract:

The Galerkin finite element solution $u_h$ of the Poisson equation $-\Delta u=f$ under the Neumann boundary condition in a possibly nonconvex polygon $\varOmega$, with a graded mesh locally refined at the corners of the domain, is shown to satisfy the following maximum-norm stability: \begin{align*} \|u_h\|_{L^{\infty }(\varOmega )} \le C\ell _h\|u\|_{L^{\infty }(\varOmega )} , \end{align*} where $\ell _h = \ln (2+1/h)$ for piecewise linear elements and $\ell _h=1$ for higher-order elements. As a result of the maximum-norm stability, the following best approximation result holds: \begin{align*} \|u-u_h\|_{L^{\infty }(\varOmega )} \le C\ell _h\|u-I_hu\|_{L^{\infty }(\varOmega )} , \end{align*} where $I_h$ denotes the Lagrange interpolation operator onto the finite element space. For a locally quasi-uniform triangulation sufficiently refined at the corners, the above best approximation property implies the following optimal-order error bound in the maximum norm: \begin{align*} \|u-u_h\|_{L^\infty (\varOmega )} \le \begin {cases} C\ell _h h^{k+2-\frac {2}{p}} \|f\|_{W^{k,p}(\varOmega )} &\text {if $r\ge k+1$}, \\ C\ell _h h^{k+1} \|f\|_{H^{k}(\varOmega )} &\text {if $r=k$}, \end{cases} \end{align*} where $r\ge 1$ is the degree of finite elements, $k$ is any nonnegative integer no larger than $r$, and $p\in [2,\infty )$ can be arbitrarily large.