Multilevel iterative solvers for the edge finite element solution of the 3D Maxwell equation

Abstract

In the edge vector finite element solution of the frequency domain Maxwell equations, the presence of a large kernel of the discrete rotor operator is known to ruin convergence of standard iterative solvers. We extend the approach of [R. Hiptmair, Multigrid method for Maxwell's equations, SIAM J. Numer. Anal. 36 (1) (1999) 204-225] and, using domain decomposition ideas, construct a multilevel iterative solver where the projection with respect to the kernel is combined with the use of a hierarchical representation of the vector finite elements.The new iterative scheme appears to be an efficient solver for the edge finite element solution of the frequency domain Maxwell equations. The solver can be seen as a variable preconditioner and, thus, accelerated by Krylov subspace techniques (e.g. GCR or FGMRES). We demonstrate the efficiency of our approach on a test problem with strong jumps in the conductivity.

Multilevel algebraic methods of modeling the 3D electromagnetic field

“…After this, the period of slow convergence of the method begins (the suppression of low-frequency component of the error). The suppression of the low-frequency component is provided by using the correction of the solution in the kernel space of the curl-operator [3] (Fig.2).…”

“…The usage of classical iterative Krylov subspaces methods with incomplete LU-factorization, its modifications or diagonal scaling is not effective for solving this type of systems of linear equations [2,3,4]. These methods converge to the solution very slowly, or even diverge.…”

“…These methods converge to the solution very slowly, or even diverge. The multilevel (multigrid) algebraic solvers, that allow taking into account the peculiarities of the problem and the kernel space of the curl-operator, are used for such class of problems [2,3].…”

The mathematical modeling of the electric field in the media with anisotropic objects

“…This condition is done automatically on interior and exterior boundaries of the domains with contrast physical properties. There are the computational schemes that provide for fulfillment of conservation low in weak form [11]. These schemes guarantee computing accuracy of the electrical field normal component jump on the interior boundary.…”

The Calculation of the Effective Tensor Coefficient of the Medium for the Objects with Microinclusions

“…The boundary conditions (2) and (3) are fulfilled in the vector F. The system of linear algebraic Equations (10) is solved by the special two-level solver [21].…”