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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| Published in: | IEEE transactions on antennas and propagation Vol. 70; no. 2; pp. 1187 - 1197 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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New York
IEEE
01.02.2022
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
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| ISSN: | 0018-926X, 1558-2221 |
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| Abstract | 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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| AbstractList | 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. |
| Author | Jiao, Dan Xue, Li |
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| Snippet | 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... 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... |
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| SubjectTerms | 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 |
| Title | Fast Method for Accelerating Convergence of Iterative Partial Differential Equation Solvers by Changing System Matrix to Laplacian Counterpart |
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