Domain Decomposition Method For Maxwell’s Equations

This work is concerned with the development of numerical methods for the simulation of time-harmonic electromagnetic wave propagation problems. It is aims at developing a high-performance numerical methodology for time-harmonic Maxwell’s equations. We usually consider finite difference, finite element, discontinuous Galerkin(DG) methods. We use a hybridizable discontinuous Galerkin(HDG) method to discretize the two-dimensional time-harmonic Maxwell’s equations on a triangular mesh. There are lots of advantages in the HDG methods. They can adapt to complex geometries and non-conforming meshes. High order accuracy, hp-adaptivity and natural parallelism can been achieved by using HDG methods. An additional hybrid variable is putted forward by HDG methods on the faces of the elements. Local solutions can be definite on those faces, then HDG methods produce a linear system in terms of the degrees of freedoms(DOFs) of the additional hybrid variable only. So the number of globally coupled DOFs is reduced associated to a classical upwind flux-based DG method.Though the HDG method results in a smaller linear system, the size of this system is often too large to be solved by a direct solver as soon as one consider realistic three-dimensional problems. We use the domain decomposition principles to solve the problem. Thanks to domain decomposition method for large-scale problems can be broken down into small problems, the complexity of boundary value problems for the decomposition of simple boundary value problem, so it has good parallel performance. Optimized Schwarz method based on the classical Schwarz methods and is a branch of the domain decomposition method(DDM). These methods use more effective transmission conditions between subdomains than the classical Dirichlet conditions.This article based on the domain decomposition methods. In each sub-domain, we use non-conforming meshes for time-harmonic Maxwell’s equations discretized by a HDG method. The selection of optimal transmission conditions in the sub-domain, you can get a better convergence effect. The characteristic of this paper is based on domain decomposition of Schwarz method and discrete the Maxwell’s equations in sub-domain level. A distinguishing feature of the present work is we show the HDG method naturally couples with a Schwarz method relying on optimized transmission conditions. This strategy based on optimized Schwarz methods applied to the time-harmonic Maxwell system discretized by a HDG method and leads to the best possible convergence of these algorithms. It is greatly reduced the number of globally coupled DOFs as well as save both CPU time and memory cost compared to a DG method.Formula derivation and theoretical analysis of optimized Schwarz method combined with HDG method for two-dimensional time-harmonic Maxwell’s equations are given in the article. The presented numerical results show the effectiveness of the optimized DDM-HDG method. The efficiency is further demonstrated on more realistic three-dimensional geometries including a lot of practical application.