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IMA Journal of Numerical Analysis Advance Access published online on August 6, 2009
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drp014
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© The author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

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

Stig Larsson{dagger} and Fardin Saedpanah{ddagger}

Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, SE–412 96 Göteborg, Sweden

{dagger} Email: stig{at}chalmers.se

{ddagger} Corresponding author. Email: fardin{at}chalmers.se

Received on 4 September 2008. Revised on 10 February 2009.


  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.

Key Words: finite element; continuous Galerkin; linear viscoelasticity; fractional calculus; weakly singular kernel; stability; a priori error estimate


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