Abstract:A fast method of an arbitrary high order for approximating volume potentials is proposed, which is effective also in high dimensional cases. Basis functions introduced in the theory of approximate approximations are used. Results of numerical experiments, which show approximation order $O(h^8)$ for the Newton potential in high dimensions, for example, for $n= 200 000$, are provided. The computation time scales linearly in the space dimension. New one-dimensional integral representations with separable integrands of the potentials of advection-diffusion and heat equations are obtained.
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- Flavia Lanzara
- Affiliation: Dipartimento di Matematica, Università “La Sapienza”, Piazzale Aldo Moro 2, 00185 Rome, Italy
- Email: email@example.com
- Vladimir Maz’ya
- Affiliation: Department of Mathematical Sciences, M&O Building, University of Liverpool, Liverpool L69 3BX, United Kingdom –and– Department of Mathematics, University of Linköping, 581 83 Linköping, Sweden
- MR Author ID: 196507
- Email: firstname.lastname@example.org
- Gunther Schmidt
- Affiliation: Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany
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- Received by editor(s): November 2, 2009
- Received by editor(s) in revised form: February 20, 2010
- Published electronically: September 22, 2010
- Additional Notes: This research was partially supported by the UK and Engineering and Physical Sciences Research Council via the grant EP/F005563/1.
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
- Journal: Math. Comp. 80 (2011), 887-904
- MSC (2010): Primary 65D32; Secondary 65-05
- DOI: https://doi.org/10.1090/S0025-5718-2010-02425-1
- MathSciNet review: 2772100