On the convergence of the combination technique

Griebel, Michael and Harbrecht, Helmut. (2014) On the convergence of the combination technique. In: Sparse Grids and Applications - Munich 2012. Berlin-Heidelberg, pp. 55-74.

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Official URL: http://edoc.unibas.ch/dok/A6254415

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Sparse tensor product spaces provide an efficient tool todiscretize higher dimensional operator equations. The direct Galerkin method in such ansatz spaces may employ hierarchical bases, interpolets, wavelets or multilevel frames. Besides, an alternative approach is provided by the so-called combination technique. It properly combines the Galerkin solutions of the underlying problem on certain full (but small) tensor product spaces. So far, however, the combination technique has been analyzed only for special model problems. In the present paper, we provide now the analysis of the combination technique for quite general operator equations in sparse tensor product spaces. We prove that the combination technique produces the same order of convergence as the Galerkin approximation with respect to the sparse tensor product space. Furthermore, the order of the cost complexity is the same as for the Galerkin approach in the sparse tensor product space. Our theoretical findings are validated by numerical experiments.
Faculties and Departments:05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Computational Mathematics (Harbrecht)
UniBasel Contributors:Harbrecht, Helmut
Item Type:Book Section, refereed
Book Section Subtype:Further Contribution in a Book
Note:Also published in: Lecture Notes in Computational Science and Engineering. - Berlin-Heidelberg : Springer, 2014. - Vol. 97 (2014), S. 55-74 -- Publication type according to Uni Basel Research Database: Book item
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Last Modified:31 Dec 2015 10:55
Deposited On:23 May 2014 08:34

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