Unified error analysis for nonconforming space discretizations of wave-type equations

Abstract

Abstract This paper provides a unified error analysis for nonconforming space discretizations of linear wave equations in the time domain. We propose a framework that studies wave equations as first-order evolution equations in Hilbert spaces and their space discretizations as differential equations in finite-dimensional Hilbert spaces. A lift operator maps the semidiscrete solution from the approximation space to the continuous space. Our main results are a priori error bounds in terms of interpolation, data and conformity errors of the method. Such error bounds are the key to the systematic derivation of convergence rates for a large class of problems. To show that this approach significantly eases the proof of new convergence rates, we apply it to an isoparametric finite element discretization of the wave equation with acoustic boundary conditions in a smooth domain. Moreover, our results reproduce known convergence rates for already investigated conforming and nonconforming space discretizations in a concise and unified way. The examples discussed in this paper comprise discontinuous Galerkin discretizations of Maxwell’s equations and finite elements with mass lumping for the acoustic wave equation.