Maximum-norm estimates for finite-element methods for a strongly damped wave equation
The article refers to earlier work of \textit{S. Larsson, V. Thomée}, and \textit{L. B. Wahlbin}, [IMA J. Numer. Anal. 11, No.~1, 115--142 (1991; Zbl 0718.65070)], which concerns 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 \(L_2\)-based norms. The analysis depended on the fact that the solution may be expressed in terms of an analytic semigroup. In this paper authors combine that 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. The error estimates in maximum norm are derived.
backward Euler and Crank-Nicolson time stepping methods
scientific article; zbMATH DE number 2081535