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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
101/2018

Numerical approximation of Poisson problems in long domains

Michel Chipot, Wolfgang Hackbusch, Stefan A. Sauter and Alexander Veit

Abstract

In this paper, we consider the Poisson equation on a "long" domain which is the Cartesian product of a one-dimensional long interval with a (d-1)-dimensional domain. The right-hand side is assumed to have a rank-1 tensor structure. We will present methods to construct approximations of the solution which have tensor structure and the computational effort is governed by only solving elliptic problems on lower-dimensional domains. A zero-th order tensor approximation is derived by using tools from asymptotic analysis (method 1). The resulting approximation is an elementary tensor and, hence has a fixed error which turns out to be very close to the best possible approximation of zero-th order. This approximation can be used as a starting guess for the derivation of higher-order tensor approximations by an alternating-least-squares (ALS) type method (method 2). Numerical experiments show that the ALS is converging towards the exact solution (although a rigorous and general theoretical framework is missing for our application).

Method 3 is based on the derivation of a tensor approximation via exponential sums applied to discretised differential operators and their inverses. It can be proved that this method converges exponentially with respect to the tensor rank. We present numerical experiments which compare the performance and sensitivity of these three methods.

Received:
Nov 29, 2018
Published:
Nov 30, 2018
MSC Codes:
15A69, 35B40, 35J2, 65K05
Keywords:
Poisson problem, long domain, asymptotic analysis, tensor approximation, alternating least squares

Related publications

inJournal
2022 Journal Open Access
Michel Chipot, Wolfgang Hackbusch, Stefan A. Sauter and Alexander Veit

Numerical approximation of Poisson problems in long domains

In: Vietnam journal of mathematics, 50 (2022) 2, pp. 375-393