Maximum-Norm Estimates for Finite-Element Methods for a Strongly Damped Wave Equation

Authors: Thomée V.¹; Wahlbin L.B.²

Source: Bit Numerical Mathematics, Volume 44, Number 1, 2004 , pp. 165-179(15)

Publisher: Springer

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Abstract:

We return to earlier work in Larsson, Thomée, and Wahlbin, IMA J. Numer. Anal. 11 (1991), concerning the numerical solution of a homogeneous linear wave equation with strong damping, arising in viscoelasticity. In that work spatial discretization by finite elements and associated fully discrete methods were analyzed in L₂-based norms. The analysis depended on the fact that the solution may be expressed in terms of an analytic semigroup. In the present work we combine this approach with recent results on discretization of parabolic problems to derive essentially optimal order error bounds in maximum-norm for piecewise linear finite elements combined with backward Euler and Crank–Nicolson time stepping methods.

Keywords: strongly damped wave equation; finite elements; error bounds; semigroup; resolvent estimate; maximum-norm

Document Type: Research article

DOI: http://dx.doi.org/10.1023/B:BITN.0000025091.78408.e4

Affiliations: 1: Department of Mathematics, Chalmers University of Technology and Göteborg University, SE-412 96 Göteborg, Sweden., Email: thomee@math.chalmers.se 2: Department of Mathematics, Cornell University, Ithaca, NY 14853, USA., Email: wahlbin@math.cornell.edu

Publication date: 2004-01-01