Analytic Regularity for Linear Elliptic Systems in Polygons and Polyhedra
Résumé
We prove weighted anisotropic analytic estimates for solutions of model elliptic boundary value problems in polyhedra. The weighted analytic classes which we use are the same as those introduced by B. Guo in 1993 in view of establishing exponential convergence for hp methods in polyhedra.
We first give a simple proof of the weighted analytic regularity in a polygon, relying on new elliptic a priori estimates with analytic control of derivatives in smooth domains. The technique is based on dyadic partitions near the corners. This technique can be successfully extended to polyhedra, but only isotropic analytic regularity can be proved in this way. We therefore combine it with a nested open set technique to obtain the three-dimensional anisotropic analytic result. Our proofs are global and do not rely on the analysis of singularities.
We first give a simple proof of the weighted analytic regularity in a polygon, relying on new elliptic a priori estimates with analytic control of derivatives in smooth domains. The technique is based on dyadic partitions near the corners. This technique can be successfully extended to polyhedra, but only isotropic analytic regularity can be proved in this way. We therefore combine it with a nested open set technique to obtain the three-dimensional anisotropic analytic result. Our proofs are global and do not rely on the analysis of singularities.
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