Maximum-norm estimates for finite-element methods for a strongly damped wave equation
We return to earlier work in Larsson, Thomee, 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 L-2-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.
maximum-norm strongly damped wave equation finite elements semigroup error bounds resolvent estimate