Constantin Bacuta, Victor Nistor, Jinchao Xu, Ludmil Zikatanov
A Note on Improving the Rate of Convergence of `High Order Finite Elements' on Polygons
Preprint series: ESI preprints

MSC:

65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods

65N50 Mesh generation and refinement

35J05 Laplace equation, reduced wave equation (Helmholtz), Poisson equation, See also {31Axx, 31Bxx}

Abstract: Let $u$ and $u_{V_n}$ be the solution and, respectively, the
finite element solution of the Poisson's equation $\Delta u = f$
with zero boundary conditions. We construct for any $m \in \NN$
and any polygon $\PP$ a sequence of finite dimensional subspaces
$V_n$ such that $\normH{u - u_{V_n}}{1} \le C
\dim(V_n)^{-m/2}\normH{f}{m-1}$, where $f \in H^{m-1}(\PP)$ is
arbitrary and $C$ is a constant that depends only on $\PP$ (we do
\emph{not} assume $u \in H^{m+1}(\PP)$). Although the final result
is in terms of the ``usual'' Sobolev spaces, the proof relies on
estimates for the Poisson problem in Sobolev spaces with weights.
Other ``$h^m$''--type approximation results are also obtained.
This is an announcement, but some sketches of the proofs of the
main results are provided. Full details of the proofs and complete
references will be provided in a different paper.
Keywords: finite element method, triangulation, meshes, Laplace's equation, Poisson's equation, boundary value problem, Sobolev space, weighted Sobolev space