Optimal \(L^{\infty}\)-error estimates for piecewise linear finite elements in \({\mathbb{R}}^ n\) for 1\(\leq n\leq 3\)
For two second order elliptic boundary value problems, uniform convergence error estimates are derived using piecewise linear (conforming and non-conforming) finite elements. The estimates are valid for polyhedral convex regions \(\Omega \subset \mathbb{R}^ n\) with \(1\leq n\leq 3\). The proof of the estimates is mainly based on Sobolev's inequality, by which the uniform estimates are reduced to estimates in the mean. In this way, it is possible to obtain the desired uniform estimate \(O(h^ 2)\) in the plane as a special case, thus conforming theoretically Natterer's numerical results.
scientific article; zbMATH DE number 3947561
scientific article