Fast Method for Accelerating Convergence of Iterative Partial Differential Equation Solvers by Changing System Matrix to Laplacian Counterpart

In this work, we find that the matrix representing the curl-curl operator in a partial differential equation solver of Maxwell's equations can be analytically decomposed into a gradient divergence operator and a Laplacian, both of which can be constructed from the mesh information without any n...

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Bibliographic Details
Published in:IEEE transactions on antennas and propagation Vol. 70; no. 2; pp. 1187 - 1197
Main Authors: Xue, Li, Jiao, Dan
Format: Journal Article
Language:English
Published: New York IEEE 01.02.2022
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Computational efficiency
Computing costs
Conductors
Convergence
Divergence
Fast convergence
fast method
finite-difference method
finite-element method
frequency domain
Frequency-domain analysis
Helmholtz decomposition
Iterative methods
Iterative solution
iterative solver
Laplace equations
Mathematical analysis
Matrix decomposition
Maxwell equations
Operators (mathematics)
partial differential equation method
Partial differential equations
Solvers
Sparse matrices
ISSN:0018-926X, 1558-2221
Online Access:Get full text
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Summary:In this work, we find that the matrix representing the curl-curl operator in a partial differential equation solver of Maxwell's equations can be analytically decomposed into a gradient divergence operator and a Laplacian, both of which can be constructed from the mesh information without any need for computation. The curl-curl operator can hence be replaced by the Laplacian to find the divergence-free component of the field solution. The Laplacian is positive definite and well-conditioned. As a result, the convergence of an iterative solution of Maxwell's equations can be guaranteed, and also significantly accelerated. Based on the finding, we represent the divergence-free component of the unknown field solution by deducting its curl-free component. The curl-free component resides in the nullspace of the curl-curl operator, which is also analytically known from the mesh information no matter it is a regular grid or an unstructured mesh. After the divergence-free component is rapidly solved from a Laplacian counterpart of the original system matrix, the curl-free component can also be solved from a Laplacian matrix, and hence having fast and guaranteed convergence. The total computational cost of the proposed method is simply a small number of sparse matrix-vector multiplications. The proposed method has been successfully applied to solve ill-conditioned on-chip, packaging, and antenna radiation problems at both low and high frequencies, involving both inhomogeneous dielectrics and lossy conductors. Numerical experiments have demonstrated its fast and guaranteed convergence, as well as trivial computational cost independent of problem size.
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ISSN:0018-926X
1558-2221
DOI:10.1109/TAP.2021.3111509