On the inversion of polynomials of discrete Laplace matrices

S. S. Asghar; Q. Peng; F.J. Vermolen; Cornelis Vuik

doi:10.1016/j.rinam.2026.100686

On the inversion of polynomials of discrete Laplace matrices

S. S. Asghar , Q. Peng, F.J. Vermolen, Cornelis Vuik

Numerical Analysis

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Abstract

The efficient inversion of matrix polynomials is a critical challenge in computational mathematics. We design a procedure to determine the inverse of matrices polynomial of multidimensional Laplace matrices. The method is based on eigenvector and eigenvalue expansions. The method is consistent with previously known expressions of the inverse discretized Laplacian in one spatial dimension (Vermolen et al., 2022). The formalism is further extended to obtain closed form expressions for time-dependent problems.

Original language	English
Article number	100686
Number of pages	15
Journal	Results in Applied Mathematics
Volume	29
DOIs	https://doi.org/10.1016/j.rinam.2026.100686
Publication status	Published - 2026

Keywords

Laplace matrix
Inverse matrix
Solution to linear systems
Eigenvector expansion

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10.1016/j.rinam.2026.100686Licence: CC BY

1-s2.0-S259003742600004X-mainFinal published version, 2 MBLicence: CC BY

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title = "On the inversion of polynomials of discrete Laplace matrices",

abstract = "The efficient inversion of matrix polynomials is a critical challenge in computational mathematics. We design a procedure to determine the inverse of matrices polynomial of multidimensional Laplace matrices. The method is based on eigenvector and eigenvalue expansions. The method is consistent with previously known expressions of the inverse discretized Laplacian in one spatial dimension (Vermolen et al., 2022). The formalism is further extended to obtain closed form expressions for time-dependent problems.",

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AU - Vuik, Cornelis

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N2 - The efficient inversion of matrix polynomials is a critical challenge in computational mathematics. We design a procedure to determine the inverse of matrices polynomial of multidimensional Laplace matrices. The method is based on eigenvector and eigenvalue expansions. The method is consistent with previously known expressions of the inverse discretized Laplacian in one spatial dimension (Vermolen et al., 2022). The formalism is further extended to obtain closed form expressions for time-dependent problems.

AB - The efficient inversion of matrix polynomials is a critical challenge in computational mathematics. We design a procedure to determine the inverse of matrices polynomial of multidimensional Laplace matrices. The method is based on eigenvector and eigenvalue expansions. The method is consistent with previously known expressions of the inverse discretized Laplacian in one spatial dimension (Vermolen et al., 2022). The formalism is further extended to obtain closed form expressions for time-dependent problems.

KW - Laplace matrix

KW - Inverse matrix

KW - Solution to linear systems

KW - Eigenvector expansion

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JO - Results in Applied Mathematics

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On the inversion of polynomials of discrete Laplace matrices

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Keywords

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