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On nonnegativity preservation in finite element methods for the heat equation with non-Dirichlet boundary conditions

Chapter in book
Authors Stig Larsson
Vidar Thomée
Published in Contemporary Computational Mathematics - A celebration of the 80th birthday of Ian Sloan. Dick, Josef, Kuo, Frances Y., Wozniakowski, Henryk (Eds.)
Pages 793-814
ISBN 978-3-319-72455-3
Publisher Springer
Publication year 2018
Published at Department of Mathematical Sciences
Pages 793-814
Language en
Keywords maximum principle, Robin boundary condition, lumped mass method, Neumann boundary condition
Subject categories Computational Mathematics, Mathematical Analysis

Abstract

By the maximum principle the solution of the homogeneous heat equation with homogeneous Dirichlet boundary conditions is nonnegative for positive time if the initial values are nonnegative. In recent work it has been shown that this does not hold for the standard spatially discrete and fully discrete piecewise linear finite element methods. However, for the corresponding semidiscrete and Backward Euler Lumped Mass methods, nonnegativity of initial data is preserved, provided the un- derlying triangulation is of Delaunay type. In this paper, we study the correspond- ing problems where the homogeneous Dirichlet boundary conditions are replaced by Neumann and Robin boundary conditions, and show similar results, sometimes requiring more refined technical arguments.

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