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The continuous Galerkin method for an integro-differential equation modeling dynamic fractional order viscoelasticity

Journal article
Authors Stig Larsson
Fardin Saedpanah
Published in IMA Journal of Numerical Analysis
Volume 30
Issue 4
Pages 964-986
ISSN 1464-3642
Publication year 2010
Published at Department of Mathematical Sciences, Mathematics
Pages 964-986
Language en
Links dx.doi.org/10.1093/imanum/drp014
imajna.oxfordjournals.org/cgi/conte...
Keywords finite element; continuous Galerkin; linear viscoelasticity; fractional calculus; weakly singular kernel; stability; a priori error estimate
Subject categories Numerical analysis

Abstract

We consider a fractional order integro-differential equation with a weakly singular convolution kernel. The equation with homogeneous mixed Dirichlet and Neumann boundary conditions is reformulated as an abstract Cauchy problem, and well-posedness is verified in the context of linear semigroup theory. Then we formulate a continuous Galerkin method for the problem, and we prove stability estimates. These are then used to prove a priori error estimates. The theory is illustrated by a numerical example.

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