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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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Dijkema, T.J., Stevenson, R. A sparse Laplacian in tensor product wavelet coordinates. Numer. Math. 115, 433–449 (2010). https://doi.org/10.1007/s00211-010-0288-5
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DOI: https://doi.org/10.1007/s00211-010-0288-5
Keywords
- Sparse representations
- Tensor product approximation
- Adaptive wavelet scheme
- Riesz bases
- Cubic Hermite splines