Rank Structured Approximation Method for Quasi-Periodic Elliptic Problems
Khoromskij, B., & Repin, S. (2017). Rank Structured Approximation Method for Quasi-Periodic Elliptic Problems. Computational Methods in Applied Mathematics, 17(3), 457-477. https://doi.org/10.1515/cmam-2017-0014
Published inComputational Methods in Applied Mathematics
© 2017 Walter de Gruyter GmbH, Berlin/Boston. Published in this repository with the kind permission of the publisher.
We consider an iteration method for solving an elliptic type boundary value problem Au=f, where a positive definite operator A is generated by a quasi-periodic structure with rapidly changing coefficients (a typical period is characterized by a small parameter ϵ). The method is based on using a simpler operator A0 (inversion of A0 is much simpler than inversion of A), which can be viewed as a preconditioner for A. We prove contraction of the iteration method and establish explicit estimates of the contraction factor q. Certainly the value of q depends on the difference between A and A0. For typical quasi-periodic structures, we establish simple relations that suggest an optimal A0 (in a selected set of “simple” structures) and compute the corresponding contraction factor. Further, this allows us to deduce fully computable two-sided a posteriori estimates able to control numerical solutions on any iteration. The method is especially efficient if the coefficients of A admit low-rank representations and if algebraic operations are performed in tensor structured formats. Under moderate assumptions the storage and solution complexity of our approach depends only weakly (merely linear-logarithmically) on the frequency parameter 1/ϵ. ...
Publication in research information system
MetadataShow full item record
Showing items with similar title or keywords.
Wolfmayr, Monika (Elsevier, 2020)In this paper, a new technique is shown for deriving computable, guaranteed lower bounds of functional type (minorants) for two different cost functionals subject to a parabolic time-periodic boundary value problem. Together ...
Guaranteed error bounds and local indicators for adaptive solvers using stabilised space-time IgA approximations to parabolic problems Langer, Ulrich; Matculevich, Svetlana; Repin, Sergey (Elsevier, 2019)The paper is concerned with space–time IgA approximations to parabolic initial–boundary value problems. We deduce guaranteed and fully computable error bounds adapted to special features of such type of approximations and ...
Guaranteed and computable error bounds for approximations constructed by an iterative decoupling of the Biot problem Kumar, Kundan; Kyas, Svetlana; Nordbotten, Jan Martin; Repin, Sergey (Elsevier, 2021)The paper is concerned with guaranteed a posteriori error estimates for a class of evolutionary problems related to poroelastic media governed by the quasi-static linear Biot equations. The system is decoupled by employing ...
Apushkinskaya, Darya E.; Repin, Sergey I. (Pleiades Publishing, 2020)The paper is concerned with an elliptic variational inequality associated with a free boundary obstacle problem for the biharmonic operator. We study the bounds of the difference between the exact solution (minimizer) of ...
Anjam, Immanuel; Pauly, Dirk (University of Jyväskylä, 2014)