Weighted analytic regularity in polyhedra
Slides of a lecture given at
The results presented in these slides are proved in the paper
''Analytic Regularity for Linear Elliptic Systems in Polygons and Polyhedra''
Martin Costabel, Monique Dauge and Serge Nicaise.
Math. Models Methods Appl. Sci. 22, 1250015 (2012) [63 pages]
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
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 $p$-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
- The solutions to standard elliptic boundary problems belong to such analytic classes.
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.
- 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.