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Numerical approximation schemes for multi-dimensional wave equations in asymmetric spaces. (English) Zbl 1311.65112
Summary: We develop finite difference numerical schemes for a model arising in multi-body structures, previously analyzed by H. Koch and the second author [Contemp. Math. 409, 55–77 (2006; Zbl 1108.35110)], constituted by two $$n$$-dimensional wave equations coupled with a $$(n-1)$$-dimensional one along a flexible interface.
That model, under suitable assumptions on the speed of propagation in each media, is well-posed in asymmetric spaces in which the regularity of solutions differs by one derivative from one medium to the other.
Here we consider a flat interface and analyze this property at a discrete level, for finite difference and mixed finite element methods on regular meshes parallel to the interface. We prove that those methods are well-posed in such asymmetric spaces uniformly with respect to the mesh-size parameters and we prove the convergence of the numerical solutions towards the continuous ones in these spaces.
In other words, these numerical methods that are well-behaved in standard energy spaces, preserve the convergence properties in these asymmetric spaces too. These results are illustrated by several numerical experiments.

##### MSC:
 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 35L05 Wave equation 35S15 Boundary value problems for PDEs with pseudodifferential operators
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