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Numerical solution of parabolic problems based on a weak space-time formulation

Journal article
Authors Stig Larsson
Matteo Molteni
Published in Computational Methods in Applied Mathematics
Volume 17
Issue 1
Pages 65–84
ISSN 1609-4840
Publication year 2017
Published at Department of Mathematical Sciences
Pages 65–84
Language en
Links dx.doi.org/10.1515/cmam-2016-0027
Keywords Inf-Sup, Space-Time, Superconvergence, Quasi-Optimality, Finite Element, Error Estimate, Petrov–Galerkin
Subject categories Numerical analysis

Abstract

We investigate a weak space-time formulation of the heat equation and its use for the construction of a numerical scheme. The formulation is based on a known weak space-time formulation, with the difference that a pointwise component of the solution, which in other works is usually neglected, is now kept. We investigate the role of such a component by first using it to obtain a pointwise bound on the solution and then deploying it to construct a numerical scheme. The scheme obtained, besides being quasi-optimal in the L2 sense, is also pointwise superconvergent in the temporal nodes. We prove a priori error estimates and we present numerical experiments to empirically support our findings.

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