A sparse Laplacian in tensor product wavelet coordinates |
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Authors: | Tammo Jan Dijkema Rob Stevenson |
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Institution: | 1. Department of Mathematics, Utrecht University, P.O. Box 80010, 3508 TA, Utrecht, The Netherlands 2. Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Plantage Muidergracht 24, 1018 TV, Amsterdam, The Netherlands
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Abstract: | We construct a wavelet basis on the unit interval with respect to which both the (infinite) mass and stiffness matrix corresponding to the one-dimensional Laplacian are (truly) sparse and boundedly
invertible. As a consequence, the (infinite) stiffness matrix corresponding to the Laplacian on the n-dimensional unit box with respect to the n-fold tensor product wavelet basis is also sparse and boundedly invertible. This greatly simplifies the implementation and
improves the quantitative properties of an adaptive wavelet scheme to solve the multi-dimensional Poisson equation. The results
extend to any second order partial differential operator with constant coefficients that defines a boundedly invertible operator. |
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