Algebraic convergence for anisotropic edge elements in polyhedral domains. (English) Zbl 1116.78020
In this paper the authors studied the approximation of solutions of the time-harmonic Maxwell equations in a three-dimensional bounded domain filled with an isotropic, homogeneous material whose magnetic permeability and electric permittivity are given by positive constants. The approximation is achieved using finite elements in the situation where is a Lipschitz polyhedron. Such a situation is very natural from the practical point of view, but is the cause of important difficulties, and even sometimes obstructions to a converging approximation. The main reason of this is the very poor regularity of solutions when has non-convex edges and corners. Because of the poor regularity of solution, the convergence rate is very low and so is lost all benefits of the use of higher degree elements. Here is used the anisotropic weighted Sobolev regularity of the solution on domain with three-dimensional edges and corners. The authors proved the discrete compactness property needed for convergence of the Maxwell eigenvalue problem.
Reviewer: Ariadna Lucia Pletea (Iaşi)
MSC:
| 78M10 | Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N25 | Numerical methods for eigenvalue problems for boundary value problems involving PDEs |
Keywords:
time-harmonic Maxwell equation; finite elements; error estimate; discrete compactness propertyReferences:
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