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Stencils with isotropic discretization error for differential operators. (English) Zbl 1098.65108

Summary: We derive stencils, i.e. difference schemes, for differential operators for which the discretization error becomes isotropic in the lowest order. We treat the Laplacian, bi-Laplacian (= biharmonic operator), and the gradient of the Laplacian both in two and three dimensions. For three dimensions \(\mathcal O(h^2)\) results are given while for two dimensions both \(\mathcal O(h^2)\) and \(\mathcal O(h^4)\) results are presented. The results are also available in electronic form as a Mathematica file. It is shown that the extra computational cost of an isotropic stencil usually is less than 20%.

MSC:

65N15 Error bounds for boundary value problems involving PDEs
65N06 Finite difference methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J40 Boundary value problems for higher-order elliptic equations

Software:

LAPACK; PDE2D
PDFBibTeX XMLCite
Full Text: DOI

References:

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