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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.

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
Full Text: DOI
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