Volume 53, pp. 522-540, 2020.

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

Marlis Hochbruck and Jan Leibold


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.

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Key words

wave equation, dynamic boundary conditions, nonconforming space discretization, error analysis, a-priori error bounds, semilinear evolution equations, operator semigroups, isoparametric finite elements

AMS subject classifications

65M12, 65M15, 65J08, 65M60

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