Abstract
For inversion of the Laplacian subject to Dirichlet boundary conditions and, more generally, for the kth power of the Laplacian subject to boundary conditions on the function and its first k − 1 derivatives in the normal coordinate, there is a sparse symmetric, well-conditioned, projection of the operator that results from an expansion in associated Legendre polynomials.
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Ierley, G.R. A Class of Sparse Spectral Operators for Inversion of Powers of the Laplacian in N Dimensions. Journal of Scientific Computing 12, 57–73 (1997). https://doi.org/10.1023/A:1025658404257
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DOI: https://doi.org/10.1023/A:1025658404257