Numerical Study of an Anisotropic Error Estimator in the $L^2(H^1)$ Norm for the Finite Element Discretization of the Wave Equation

Abstract

An anisotropic a posteriori error estimate is derived for a finite element discretization of the wave equation in two space dimensions. Only the error due to space discretization is considered, and the error estimates are derived in the nonnatural $L^2(0,T;H^1(\Omega))$ norm using elliptic reconstruction. A numerical study of the effectivity index on unstructured, nonadapted, anisotropic meshes confirms the sharpness of the error estimator, provided the error due to time discretization is negligible compared to the finite element error. An anisotropic, adaptive finite element algorithm is then presented to control the finite element error in the $L^2(0,T;H^1(\Omega))$ norm. Numerical results on adapted meshes indicate that the error estimator slightly underestimates the true error. We conjecture that the missing information corresponds to the interpolation error between successive meshes. It is observed that the error estimator becomes sharp again when considering the damped wave equation \[\varepsilon\frac{\partial^{2}u}{\partial t^{2}}+\frac{\partial u}{\partial t}-\Delta u=f,\] with small values of $\varepsilon$, that is, when the parabolic character of the PDE becomes predominant.