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Volume 10, Issue 3
Error Splitting Preservation for High Order Finite Difference Schemes in the Combination Technique

Christian Hendricks, Matthias Ehrhardt & Michael Günther

DOI: 10.4208/nmtma.2017.m1511

Numer. Math. Theor. Meth. Appl., 10 (2017), pp. 689-710.

Published online: 2017-10

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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.

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@Article{NMTMA-10-689, author = {Christian Hendricks, Matthias Ehrhardt and Michael Günther}, title = {Error Splitting Preservation for High Order Finite Difference Schemes in the Combination Technique}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2017}, volume = {10}, number = {3}, pages = {689--710}, 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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2017.m1511}, url = {http://global-sci.org/intro/article_detail/nmtma/12365.html} }
TY - JOUR T1 - Error Splitting Preservation for High Order Finite Difference Schemes in the Combination Technique AU - Christian Hendricks, Matthias Ehrhardt & Michael Günther JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 689 EP - 710 PY - 2017 DA - 2017/10 SN - 10 DO - http://doi.org/10.4208/nmtma.2017.m1511 UR - https://global-sci.org/intro/article_detail/nmtma/12365.html KW - AB -

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.

Christian Hendricks, Matthias Ehrhardt and Michael Günther. (2017). Error Splitting Preservation for High Order Finite Difference Schemes in the Combination Technique. Numerical Mathematics: Theory, Methods and Applications. 10 (3). 689-710. doi:10.4208/nmtma.2017.m1511
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