The approximation of solutions of elliptic problems such as Laplace or Lamé problems in non-smooth domains (polygons or polyhedra) is considerably improved by some a priori adaptation to the geometrical singularities of the domain. Such adaptation can be the use of anisotropically refined meshes, in the framework of both the h and the hp version of finite elements.
For the computation of electromagnetic fields solution of Maxwell equation, such a refinement strategy may be ineffective or, even, misleading.
In order to work efficiently, one has to combine such a mesh design with a particular strategy:
(1) Either the introduction of special families of elements (the edge elements)
(2) Or the definition of a special regularization procedure by a divergence part adapted to the geometry of the domain (the regularization with weight).
The former method is classical now, but its full interaction with anisotropic elements was recently studied [1]. The later method is newer [2] and applies to polygonal domains as well as to polyhedral domains. It allows the use standard finite elements for the approximation of the solution. When the $hp$ version of finite elements is used, we prove the exponential convergence under quite general assumptions on the design of the finite element spaces [3].
| [1] | A. BUFFA, M. COSTABEL, M. DAUGE.
Algebraic convergence for anisotropic edge elements in polyhedral domains. |
| [2] | M. COSTABEL, M. DAUGE.
Weighted regularization of Maxwell equations in polyhedral domains. Numer. Math. 93 (2002), pp.~239--277. |
| [3] | M. COSTABEL, M. DAUGE, C. SCHWAB.
Exponential Convergence of the $hp$-FEM for the Weighted Regularization of Maxwell Equations in Polygonal Domains. To appear soon on the web. |
Slides of lectures given in Bad Honnef, Germany, (Sept. 2003) and Tbilisi, Georgia (Oct. 2003)
The file is located on the homepage of the
303. WE-Heraeus-Seminar on