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Regular polynomial interpolation and approximation of global solutions of linear partial differential equations

Please always quote using this URN:urn:nbn:de:0296-matheon-4431
  • We consider regular polynomial interpolation algorithms on recursively defined sets of interpolation points which approximate global solutions of arbitrary well-posed systems of linear partial differential equations. Convergence of the "limit" of the recursively constructed family of polynomials to the solution and error estimates are obtained from a priori estimates for some standard classes of linear partial differential equations, i.e. elliptic and hyperbolic equations. Another variation of the algorithm allows to construct polynomial interpolations which preserve systems of linear partial differential equations at the interpolation points. We show how this can be applied in order to compute higher order terms of WKB-approximations of fundamental solutions of a large class of linear parabolic equations. The error estimates are sensitive to the regularity of the solution. Our method is compatible with recent developments for solution of higher dimensional partial differential equations, i.e. (adaptive) sparse grids, and weighted Monte-Carlo, and has obvious applications to mathematical finance and physics.

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Metadaten
Author:Joerg Kampen
URN:urn:nbn:de:0296-matheon-4431
Referee:Peter Imkeller
Document Type:Preprint, Research Center Matheon
Language:English
Date of first Publication:2008/01/22
Release Date:2008/01/16
Tag:
Institute:Weierstraß-Institut für Angewandte Analysis und Stochastik (WIAS)
Preprint Number:428
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