Numerical approximation schemes for multi-dimensional wave equations in asymmetric spaces

Lescarret, Vincent; Zuazua, Enrique

doi:10.1090/S0025-5718-2014-02887-1

Numerical approximation schemes for multi-dimensional wave equations in asymmetric spaces
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by Vincent Lescarret and Enrique Zuazua PDF

Math. Comp. 84 (2015), 119-152 Request permission

Abstract:

We develop finite difference numerical schemes for a model arising in multi-body structures, previously analyzed by H. Koch and E. Zuazua, 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.

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