Application of a two-step preconditioning strategy to the finite element analysis for electromagnetic problems

Abstract

A two-step preconditioning strategy is presented for the conjugate gradient (CG) iterative method to solve a large system of linear equations resulting from the use of edge-based finite-element discretizations of Helmholtz equations. The key idea is to combine both the factorized sparse approximate inverse (FSAI) and the symmetric successive overrelaxation (SSOR) preconditioning techniques in two successive steps in order to obtain a better preconditioner for the original matrix equations. The newly constructed preconditioner combines the advantages of both the FSAI and SSOR preconditioners with less computational complexity without the breakdowns of incomplete factorization INTRODUCTIONThe finite-element method (FEM) is one of the most effective and versatile techniques which has been widely applied to electromagnetic-field problems. Its popularity in modeling microwave devices lies in its ability to deal with complex geometries and materials of any composition. Systematic description for the application of this method in electromagnetic can be found [1,2]. The application of the FEM to electromagnetic problems often yields a large, sparse, complex symmetric system of linear equations. These highly sparse linear equations can be solved using efficient solution techniques for sparse matrices based on either direct methods or iterative methods. Direct methods have the advantage that multiple right-hand sides can be treated efficiently. However, storing the factorized matrix is memory intensive for large matrices. Classical direct method includes Gaussian elimination method and the closely related LU decomposition with O(N 3 ) computational complexity. Moreover, these so-called "direct" methods bring "fill-in," that is, nonzero entries are created in certain positions where the coefficient matrix originally has zeros. Fill-in is undesirable because it increases both the computing time and the storage requirement. Direct solvers usually suffer from fill-in to an extent that these large problems cannot be solved at a reasonable cost, even on the state-of-the-art parallel machines. This necessitates the use of iterative algorithms instead of direct solvers to preserve the sparsity of the finite-element matrix. Especially attractive are iterative methods that involve the coefficient matrices only in terms of matrix-vector multiplication. The most powerful iterative algorithm of these types is the conjugate gradient algorithm for solving positive definite linear system [3]. The convergence rate of the CG method is mainly determined by the condition number of the coefficient matrix, which is closely related to the distribution of the eigenvalues of the coefficient matrix [4]. The condition number of a linear system of equations usually increases with the number of unknowns. Therefore, the solution of the linear system is the main expense of the whole problem solution, which consumes more than 95% of the CPU time for FEM application in electromagnetics. It is then desirable to precondition the coefficient matrix...

Extend two-step preconditioning technique for the Crank-Nicolson finite-difference time-domain method to analyze the 3D microwave circuits

“…The efficient SSOR preconditioning strategy deployed in this work follows the implementation described in [12]. A two-step preconditioner [13,14] for the GMRES method is also proposed to solve the large sparse linear matrix equations from implicit CN-FDTD scheme in this article. As said earlier, the SAI is applied to the original matrix equation in which a banded sparsity pattern is introduced in the first step.…”

“…In this article, a new preconditioner for the GMRES method is constructed, which is based on a two-step preconditioning strategy [13,14]. This preconditioning strategy is combined with both the sparse approximate inverse (SAI) [9,10] and the symmetric successive over-relaxation (SSOR) [11,12] preconditioning techniques in two successive steps to obtain a better preconditioner for the original matrix equations.…”

Spectral Two-Step Preconditioning of Multilevel Fast Multipole Algorithm for the Fast Monostatic RCS Calculation

“…Inspired by the two-step preconditioning idea in reference [11], a spectral preconditioner proposed in [8] can be introduced and used in a two-step way for the above SAI preconditioned system. The purpose here is to recover global information by removing the effect of some smallest eigenvalues in magnitude in the SAI preconditioned matrix, which potentially can slow down the convergence of Krylov solvers [12].…”