## Weighted analytic regularity in polyhedra

Monique Dauge

Slides of a lecture given at

High-Order Numerical Approximation for Partial Differential Equations
Mathematik-Zentrum, Lipschitz Lecture Hall, 6-10 February 2012.

The results presented in these slides are proved in the paper

''Analytic Regularity for Linear Elliptic Systems in Polygons and Polyhedra''
by Martin Costabel, Monique Dauge and Serge Nicaise.
Math. Models Methods Appl. Sci. 22, 1250015 (2012) [63 pages] DOI: 10.1142/S0218202512500157

On open-archives HAL and arXiv

Abstract

Elliptic boundary value problems with analytic data (coefficients, domain and right hand sides) are analytic up to the boundary, Morrey-Nirenberg, 1957. Such a regularity allows exponential convergence of the $p$-version of the finite element method. This result does not hold in the same form in domains with edges and corners.

In ca. 1988 Babuska and Guo started a program relying on three ideas
• The $p$-version should be replaced by the $hp$-version of finite elements,
• The $hp$-version is exponentially converging if solutions belong to certain weighted analytic classes (which they call countably normed spaces),
• The solutions to standard elliptic boundary problems belong to such analytic classes.
They proved results covering these three points for bi-dimensional problems (Laplace, Lamé). They started the investigation of three-dimensional problems in several papers published in 1995 and 1997. But three-dimensional problems have a level of complexity higher than 2D, for two reasons
• The $hp$-version requires anisotropic refinement along edges to prevent a blowing up of the number of degrees of freedom. Exponential convergence with such discretization will be obtain only if solutions belong to anisotropic analytic weighted spaces.
• The edge behavior combine in a non-trivial way at each corner.
We report on our result of anisotropic weighted regularity for a full class of coercive variational problems, homogeneous with constant coefficients. This completes for the first time point 3. of the Babuska-Guo program in polyhedra. The cases of Laplace-Dirichlet and Laplace-Neumann problems serve as threads along our way in the various possible combinations of weights in two- and three-dimensional domains, and the various corresponding statements. General statements are provided as well.