A posteriori error estimates for a coupled wave system with a local damping. (English. Russian original) Zbl 1283.65089
J. Math. Sci., New York 175, No. 3, 228-248 (2011); translation from Probl. Mat. Anal. 56, 17-34 (2011).
Summary: We study a finite element method applied to a system of coupled wave equations in a bounded smooth domain in \(\mathbb R^d\), \(d=1,2,3,\) associated with a locally distributed damping function. We start with a spatially continuous finite element formulation allowing jump discontinuities in time. This approach yields, \(L_2(L_2)\) and \(L_\infty(L_2)\), a posteriori error estimates in terms of weighted residuals of the system. The proof of the a posteriori error estimates is based on the strong stability estimates for the corresponding adjoint equations. Optimal convergence rates are derived upon the maximal available regularity of the exact solution and justified through numerical examples.
MSC:
| 65M15 | Error bounds 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 |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 35L05 | Wave equation |
Keywords:
finite element method; wave equation; a posteriori error estimate; stability; convergence; numerical exampleReferences:
| [1] | E. Burman, ”Adaptive finite element methods for compressible two-phase flow,” Math. Models Methods Appl. Sci. 10, No. 7, 963–989 (2000). · Zbl 1018.76024 |
| [2] | R. Codina, ”Finite element approximation of the hyperbolic wave equation in mixed form,” Comput. Methods Appl. Mech. Eng. 197, No. 13–16, 1305–1322 (2008). · Zbl 1162.65388 · doi:10.1016/j.cma.2007.11.006 |
| [3] | M. Ainsworth and T. J. Oden, A Posteriori Error Estimation in Finite Element Analysis, Wiley, Chichester (2000). · Zbl 1008.65076 |
| [4] | M. Asadzadeh, ”A posteriori error estimates for the Fokker–Planck and Fermi pencil beam equations,” Math. Models Methods Appl. Sci. 10, No. 5, 737–769 (2000). · Zbl 1012.82008 |
| [5] | M. Asadzadeh and P. Kowalczyk, ”Convergence analysis of the streamline diffusion and discontinuous Galerkin methods for the Vlasov–Fokker–Planck system,” Numer. Methods Partial Differ. Equations 21, No. 3, 472–495 (2005). · Zbl 1072.82023 · doi:10.1002/num.20044 |
| [6] | E. H. Gergoulus, O. Lakkis, and C. Makridakis, ”A posteriori L L 2)-error bounds in finite element approximation of the wave equation,” arXiv:1003.3641v1 [math.NA] (2010). |
| [7] | T. J. R. Hughes and A. N. Brooks, ”A theoretical framework for Petrov–Galerkin methods with discontinuous weighting functions: Application to the streamline upwind procedure,” in: Finite Elements in Fluids, Vol. IV, Wiley, Chichester, pp. 46–65 (1982). |
| [8] | M. Izadi, ”A posteriori error estimates for the coupling equations of scalar conservation laws,” BIT. Numer. Math. 49, No. 4, 697–720 (2009). · Zbl 1190.65138 · doi:10.1007/s10543-009-0243-y |
| [9] | C. Johnson and A. Szepessy, ”Adaptive finite element methods for conservation laws based on a posteriori error estimates,” Commun. Pure Appl. Math. 48, No. 3, 199–234 (1995). · Zbl 0826.65088 · doi:10.1002/cpa.3160480302 |
| [10] | C. Johnson, ”Discontinuous Galerkin finite element methods for second order hyperbolic problems,” Comput. Methods Appl. Mech. Eng. 107, No. 1–2, 117–129 (1993). · Zbl 0787.65070 · doi:10.1016/0045-7825(93)90170-3 |
| [11] | E. Godlewski and P. A. Raviart, ”The numerical interface coupling of nonlinear hyperbolic systems of conservation laws. I: The scalar case,” Numer. Math. 97, No. 1, 81–130 (2004). · Zbl 1063.65080 · doi:10.1007/s00211-002-0438-5 |
| [12] | C. A. Raposo and W. D. Bastos, ”Energy decay for the solutions of a coupled wave system,” TEMA Tend. Mat. Apl. Comput. 10, No. 2, 203–209 (2009). · Zbl 1208.35087 |
| [13] | V. Komornik and R. Bopeng, ”Boundary stabilization of compactly coupled wave equations,” Asymptotic Anal. 14, No. 4, 339–359 (1997). · Zbl 0887.35088 |
| [14] | F. Dubois and P. Le Floch, ”Boundary conditions for nonlinear hyperbolic systems of conservation laws,” J. Differ. Equations 71, No. 1, 93–122 (1988). · Zbl 0649.35057 · doi:10.1016/0022-0396(88)90040-X |
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