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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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[1] Castro, C., Asymptotic analysis and control of a hybrid system composed by two vibrating strings connected by a point mass, ESAIM Control Optim. Calc. Var., 2, 231-280 (electronic) (1997) · Zbl 1055.74543
[2] Castro, Carlos; Micu, Sorin, Boundary controllability of a linear semi-discrete 1-D wave equation derived from a mixed finite element method, Numer. Math., 102, 3, 413-462 (2006) · Zbl 1102.93004
[3] Castro, Carlos; Micu, Sorin; M{\"u}nch, Arnaud, Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square, IMA J. Numer. Anal., 28, 1, 186-214 (2008) · Zbl 1139.93005
[4] Castro, Carlos; Zuazua, Enrique, Analyse spectrale et contr\^ole d’un syst\`“eme hybride compos\'”e de deux poutres connect\'ees par une masse ponctuelle, C. R. Acad. Sci. Paris S\'er. I Math., 322, 4, 351-356 (1996) · Zbl 0846.73044
[5] Castro, Carlos; Zuazua, Enrique, A hybrid system consisting of two flexible beams connected by a point mass: spectral analysis and well-posedness in asymmetric spaces. \'Elasticit\'e, visco\'elasticit\'e et contr\^ole optimal (Lyon, 1995), ESAIM Proc. 2, 17-54 (electronic) (1997), Soc. Math. Appl. Indust.: Paris:Soc. Math. Appl. Indust. · Zbl 0898.35055
[6] Castro, Carlos; Zuazua, Enrique, Boundary controllability of a hybrid system consisting in two flexible beams connected by a point mass, SIAM J. Control Optim., 36, 5, 1576-1595 (electronic) (1998) · Zbl 0909.35085
[7] Castro, C.; Zuazua, E., Exact boundary controllability of two Euler-Bernoulli beams connected by a point mass, Math. Comput. Modelling, 32, 9, 955-969 (2000) · Zbl 0969.00019
[8] Ervedoza, Sylvain, Observability properties of a semi-discrete 1D wave equation derived from a mixed finite element method on nonuniform meshes, ESAIM Control Optim. Calc. Var., 16, 2, 298-326 (2010) · Zbl 1192.35109
[9] [G] E. H. Gaylord, C. N. Gaylord and J. E. Stallmeyer, Structural Engineering Handbook, McGraw-Hill, (1997).
[10] Hansen, Scott; Zuazua, Enrique, Exact controllability and stabilization of a vibrating string with an interior point mass, SIAM J. Control Optim., 33, 5, 1357-1391 (1995) · Zbl 0853.93018
[11] Holmer, Justin; Marzuola, Jeremy; Zworski, Maciej, Fast soliton scattering by delta impurities, Comm. Math. Phys., 274, 1, 187-216 (2007) · Zbl 1126.35068
[12] [JB] E. B\'ecache, Sch\'emas num\'eriques pour la r\'esolution de l’\'equation des ondes https://who. rocq.inria.fr/Eliane.Becache/COURS-ONDES/cours.html
[13] Koch, Herbert; Zuazua, Enrique, A hybrid system of PDE’s arising in multi-structure interaction: coupling of wave equations in \(n\) and \(n-1\) space dimensions. Recent trends in partial differential equations, Contemp. Math. 409, 55-77 (2006), Amer. Math. Soc.: Providence, RI:Amer. Math. Soc. · Zbl 1108.35110
[14] Lagnese, J. E.; Leugering, G{\`“u}nter; Schmidt, E. J. P. G., Modeling, Analysis and Control of Dynamic Elastic Multi-link Structures, Systems & Control: Foundations & Applications, xvi+388 pp. (1994), Birkh\'”auser Boston Inc.: Boston, MA:Birkh\'”auser Boston Inc. · Zbl 0810.73004
[15] [LLR] J. Le Rousseau, M. Leautaud and L. Robiano, Controllability of a parabolic system with a diffusive interface, J. Eur. Math. Soc, to appear.
[16] Lions, J.-L., Contr\^olabilit\'e Exacte, Perturbations et Stabilisation de Syst\`“emes Distribu\'”es. Tome 1, Recherches en Math\'ematiques Appliqu\'ees [Research in Applied Mathematics] 8, x+541 pp. (1988), Masson: Paris:Masson · Zbl 0653.93002
[17] Trefethen, Lloyd N., Group velocity in finite difference schemes, SIAM Rev., 24, 2, 113-136 (1982) · Zbl 0487.65055
[18] Tcheugou{\'e} T{\'e}bou, Louis Roder; Zuazua, Enrique, Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity, Numer. Math., 95, 3, 563-598 (2003) · Zbl 1033.65080
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