On the multi-level splitting of finite element spaces
Numerische Mathematik
A new family of mixed finite elements in IR3
Numerische Mathematik
Mixed and hybrid finite element methods
Mixed and hybrid finite element methods
A mixed method for approximating Maxwell's equations
SIAM Journal on Numerical Analysis
Analysis of a finite element method for Maxwell's equations
SIAM Journal on Numerical Analysis
Analysis of three-dimensional electromagnetic fields using edge elements
Journal of Computational Physics
Computing
Error Analysis of Krylov Methods In a Nutshell
SIAM Journal on Scientific Computing
Multigrid Method for Maxwell's Equations
SIAM Journal on Numerical Analysis
On the robustness of the BPX-preconditioner with respect to jumps in the coeffiencients
Mathematics of Computation
A singular field method for the solution of Maxwell's equations in polyhedral domains
SIAM Journal on Applied Mathematics
Canonical construction of finite elements
Mathematics of Computation
A justification of eddy currents model for the Maxwell equations
SIAM Journal on Applied Mathematics
Discrete compactness and the approximation of Maxwell's equations in R3
Mathematics of Computation
Finite Element Method for Elliptic Problems
Finite Element Method for Elliptic Problems
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In papers by Arnold, Falk, and Winther, and by Hiptmair, novel multigrid methods for discrete H(curl; Ω)-elliptic boundary value problems have been proposed. Such problems frequently occur in computational electromagnetism, particularly in the context of eddy current simulation.This paper focuses on the analysis of those nodal multilevel decompositions of the spaces of edge finite elements that form the foundation of the multigrid methods. It provides a significant extension of the existing theory to the case of locally vanishing coefficients and nonconvex domains. In particular, asymptotically uniform convergence of the multigrid method with respect to the number of refinement levels can be established under assumptions that are satisfied in realistic settings for eddy current problems.The principal idea is to use approximate Helmholtz-decompositions of the function space H(curl; Ω) into an H1 (Ω)-regular subspace and gradients. The main results of standard multilevel theory for H1 (Ω)-elliptic problems can then be applied to both subspaces. This yields preliminary decompositions still outside the edge element spaces. Judicious alterations can cure this.