Trivariate Spectral Collocation Approach for the Numerical Solution of Three-Dimensional Elliptic Partial Differential Equations

Musawenkhosi Patson Mkhatshwa and Melusi Khumalo
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Musawenkhosi Patson Mkhatshwa: Department of Mathematical Sciences, University of South Africa, Cnr Christian de Wet Rd. & Pioneer Avenue, Florida Park, Roodepoort 1709, South Africa
Melusi Khumalo: Department of Mathematical Sciences, University of South Africa, Cnr Christian de Wet Rd. & Pioneer Avenue, Florida Park, Roodepoort 1709, South Africa

Mathematics, 2022, vol. 10, issue 13, 1-23

Abstract: This article is concerned with the numerical solution of three-dimensional elliptic partial differential equations (PDEs) using the trivariate spectral collocation approach based on the Kronecker tensor product. By using the quasilinearization method, the nonlinear elliptic PDEs are simplified to a linear system of algebraic equations that can be discretized using the spectral collocation method. The method is based on approximating the solutions using the triple Lagrange interpolating polynomials, which interpolate the unknown functions at selected Chebyshev–Gauss–Lobatto (CGL) grid points. The CGL points are preferred to ensure simplicity in the conversion of continuous derivatives to discrete derivatives at the collocation points. The collocation process is carried out at the interior points to reduce the size of differentiation matrices. This work is aimed at verifying that the algorithm based on the method is simple and easily implemented in any scientific software to produce more accurate and stable results. The effectiveness and spectral accuracy of the numerical algorithm is checked through the determination and analysis of errors, condition numbers and computational time for various classes of single or system of elliptic PDEs including those with singular behavior. The communicated results indicate that the proposed method is more accurate, stable and effective for solving elliptic PDEs. This good accuracy becomes possible with the usage of few grid points and less memory requirements for numerical computation.

Keywords: spectral method; Chebyshev–Gauss–Lobatto points; trivariate Lagrange interpolating polynomial; quasilinearization method; three-dimensional elliptic PDEs; Kronecker tensor product (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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