Session B6 - Multiresolution and Adaptivity in Numerical PDEs
July 13, 14:30 ~ 14:55 - Room T1
Local estimates for the discrete (p-)harmonic functions for fully adaptive meshes
University of Bielefeld, Germany - email@example.com
It is well known that harmonic functions and p-harmonic functions have higher interior regularity. In 1957 De~Giorgi introduced a new technique that allows for example to estimate the maximum of the solution on a ball by an mean integral of the solution on an enlarged ball. A similar result holds for $p$-harmonic functions. The proof is based on subtle Cacciopoli estimates using truncation operators. In this talk we present similar estimates for discretely harmonic and $p$-harmonic functions. Our solutions can be scalar valued as well as vector valued, which makes a big difference for~$p$-harmonic functions.
Various results already exist in this direction for harmonic functions (e.g. Schatz-Wahlbin '95). However, the main obstacle in this direction even in the linear case is adaptivity. All of the results obtained so far, require that the mesh size does not vary too much locally. Our approach differs in such that we allow for arbitrary highly graded meshes (still shape regular). However, our approach uses certain properties of the Lagrange basis functions. This restrict our approach at the moment to acute meshes and linear elements. The proof of our result is based on a discretized version of the De~Giorgi technique.
Joint work with Toni Scharle (Oxford University, Great Britain).