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Error Splitting Preservation for High Order Finite Difference Schemes in the Combination Technique

Part of: Partial differential equations, boundary value problems Numerical approximation and computational geometry

Published online by Cambridge University Press:  20 June 2017

Christian Hendricks ,
Matthias Ehrhardt  and
Michael Günther
Christian Hendricks*
Affiliation:
Bergische Universität Wuppertal, Applied Mathematics and Numerical Analysis (AMNA), Gaußstrasse 20, 42119 Wuppertal, Germany
Matthias Ehrhardt*
Affiliation:
Bergische Universität Wuppertal, Applied Mathematics and Numerical Analysis (AMNA), Gaußstrasse 20, 42119 Wuppertal, Germany
Michael Günther*
Affiliation:
Bergische Universität Wuppertal, Applied Mathematics and Numerical Analysis (AMNA), Gaußstrasse 20, 42119 Wuppertal, Germany
*
*Corresponding author. Email addresses:christian.hendricks@gmx.de (C. Hendricks) ehrhardt@math.uni-wuppertal.de (M. Ehrhardt) guenther@math.uni-wuppertal.de (M. Günther)
*Corresponding author. Email addresses:christian.hendricks@gmx.de (C. Hendricks) ehrhardt@math.uni-wuppertal.de (M. Ehrhardt) guenther@math.uni-wuppertal.de (M. Günther)
*Corresponding author. Email addresses:christian.hendricks@gmx.de (C. Hendricks) ehrhardt@math.uni-wuppertal.de (M. Ehrhardt) guenther@math.uni-wuppertal.de (M. Günther)
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Abstract

In this paper we introduce high dimensional tensor product interpolation for the combination technique. In order to compute the sparse grid solution, the discrete numerical subsolutions have to be extended by interpolation. If unsuitable interpolation techniques are used, the rate of convergence is deteriorated. We derive the necessary framework to preserve the error structure of high order finite difference solutions of elliptic partial differential equations within the combination technique framework. This strategy enables us to obtain high order sparse grid solutions on the full grid. As exemplifications for the case of order four we illustrate our theoretical results by two test examples with up to four dimensions.

Keywords

Combination techniquesparse gridsmultivariate interpolationfinite difference schemes

MSC classification

Secondary: 65N06: Finite difference methods 65D05: Interpolation
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 3 , August 2017 , pp. 689 - 710
Copyright
Copyright © Global-Science Press 2017 

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