Maximum norm error bounds for the full discretization of nonautonomous wave equations

Abstract

In the present paper, we consider a specific class of nonautonomous wave equations on a smooth, bounded domain and their discretization in space by isoparametric finite elements and in time by the implicit Euler method. Building upon the work of Baker and Dougalis (1980, On the ${L}^{\infty }$-convergence of Galerkin approximations for second-order hyperbolic equations. Math. Comp., 34, 401–424), we prove optimal error bounds in the $W^{1,\infty } \times L^\infty $-norm for the semidiscretization in space and the full discretization. The key tool is the gain of integrability coming from the inverse of the discretized differential operator. For this, we have to pay with (discrete) time derivatives on the error in the $H^{1} \times L^2$-norm, which are reduced to estimates of the differentiated initial errors. To confirm our theoretical findings, we also present numerical experiments.