Finite element discretization of semilinear acoustic wave equations with kinetic boundary conditions

Abstract

We consider isoparametric finite element discretizations of semilinear acoustic wave equations with kinetic boundary conditions and derive a corresponding error bound as our main result. The difficulty is that such problems are stated on domains with curved boundaries and this renders the discretizations nonconforming. Our approach is to provide a unified error analysis for nonconforming space discretizations for semilinear wave equations. In particular, we introduce a general, abstract framework for nonconforming space discretizations in which we derive a-priori error bounds in terms of interpolation, data, and conformity errors. The theory applies to a large class of problems and discretizations that fit into the abstract framework. The error bound for wave equations with kinetic boundary conditions is obtained from the general theory by inserting known interpolation and geometric error bounds into the abstract error result of the unified error analysis.

An implicit–explicit time discretization scheme for second-order semilinear wave equations with application to dynamic boundary conditions

“…The wellposedness of ( 6) can be obtained by classical semigroup theory applied to the first-order formulation (7) of the equation, as detailed in [13].…”

“…In this section we combine the IMEX scheme with an abstract space discretization to obtain a fully discrete scheme. We use the framework introduced in [12] for linear equations and extended in [13] to the semilinear case. It is rather general and allows one to cover conforming as well es nonconforming space discretizations, the latter being relevant for the discretization of equations with dynamic boundary conditions.…”

“…We first introduce some notation that is required for the unified error analysis presented in [12,13].…”

“…We combine this IMEX scheme with an abstract, nonconforming space discretization within the framework of [11][12][13]. These papers provide a unified error analysis (UEA) which allows one to analyze nonconforming space discretizations of wave equations in a systematic way.…”

“…As an application of our abstract theory, we consider an acoustic wave equation with kinetic boundary conditions that fits into the abstract setting, cf., [13]. Kinetic boundary conditions are a special case of dynamic boundary conditions that contain tangential derivatives and are intrinsically posed on domains with (piecewise) smooth and therefore possibly curved boundaries.…”