A direct hierarchical multilevel preconditioner for the solution of finite element-boundary integral equations

Abstract

Boundary integral (BI) equations in combination with fast solvers such as the multilevel fast multipole method are very well suited for solving electromagnetic scattering and radiation problems. In order to consider dielectric objects, the BI approach can be extended to the hybrid finite element-boundary integral (FE-BI) method. By using hierarchical higher order basis functions for the expansion of the unknowns, very accurate results can be obtained. To accelerate the convergence of iterative solvers, the usage of preconditioning methods is indispensable. In this work, a very efficient multilevel preconditioner is presented which is based on a factorization of the submatrices containing the interactions of the zeroth order divergence-conforming basis functions in the BI formulation and the corresponding lowest order curl-conforming basis of the FE method. Moreover, it is shown that the number of fill-ins can be effectively reduced by exploiting advanced reordering algorithms.Index Terms-finite element method, boundary integral method, higher order, preconditioning. I. INTRODUCTIONBoundary integral (BI) equation methods are a versatile tool for the solution of electromagnetic scattering and radiation problems since they require only a surface discretization and the radiation condition is fulfilled automatically. While they are best suited if applied on metallic objects, dielectric materials can be easily incorporated by using a hybrid finite element-boundary integral (FE-BI) formulation [1]. In contrast to the finite element method, which is a local technique that results in a sparse matrix, the BI formulation leads to a fully populated system matrix due to the global nature of the integral operator. However, the usage of fast solvers like the multilevel fast multipole method (MLFMM) [2] reduces the complexity of the BI method considerably, so that the combined FE-BI system can be conveniently solved by iterative solution methods. If higher order basis functions [3] are deployed for the expansion of the unknowns, very accurate results can be obtained in a very efficient way for a wide range of problems.One major drawback of the depicted method is the lack of robustness due to possible convergence problems of the iterative solver. The usage of higher order hierarchical basis functions typically increases the condition number of the system matrix, thereby deteriorating the convergence. Even though advanced preconditioning strategies have been developed [4], [5], in particular to improve the conditioning of the BI matrix, for complex and/or large geometries, iterative solution techniques are still prone to fail.

A Calderón multiplicative preconditioner for the electromagnetic Poincaré–Steklov operator of a heterogeneous domain with scattering applications

“…A wide class of preconditioners for hybrid FEM-BEM formulations exists [24][25][26][27][28], with an important subset of them relying on domain decomposition methods [29,30]. For the first time, a Calderón multiplicative preconditioner is applied to a reduced hybrid FEM-BEM formulation for electromagnetic scattering at a heterogeneous obstacle, and is compatible with existing matrix vector product acceleration schemes, such as the multilevel fast multipole algorithm (MLFMA) [31][32][33][34].…”