Abstract
A wide class of well-posed operator equations can be solved in optimal computational complexity by adaptive wavelet methods. A quantitative bottleneck is the approximate evaluation of the arising residuals that steer the adaptive refinements. In this paper, we consider multi-tree approximations from tensor product wavelet bases for solving linear PDE’s. In this setting, we develop a new efficient approximate residual evaluation. Other than the commonly applied method, that uses the so-called APPLY routine, our approximate residual depends affinely on the current approximation of the solution. Our findings are illustrated by numerical results that show a considerable speed-up.
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Acknowledgments
The authors are grateful to Andreas Rupp from the Institute for Numerical Mathematics at the University of Ulm for providing implementations of \(L_2\)-orthonormal multi-wavelets on the interval.
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S.K. was supported by the Deutsche Forschungsgemeinschaft within the Research Training Group (Graduiertenkolleg) GrK 1100 Modellierung, Analyse und Simulation in der Wirtschaftsmathematik at the University of Ulm.
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Kestler, S., Stevenson, R. An Efficient Approximate Residual Evaluation in the Adaptive Tensor Product Wavelet Method. J Sci Comput 57, 439–463 (2013). https://doi.org/10.1007/s10915-013-9712-1
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DOI: https://doi.org/10.1007/s10915-013-9712-1