Super-convergence and post-processing for mixed finite element approximations of the wave equation
- 33 Downloads
Abstract
We consider the numerical approximation of acoustic wave propagation problems by mixed \(\text {BDM}_{k+1}\)–\(\text {P}_k\) finite elements on unstructured meshes. Optimal convergence of the discrete velocity and super-convergence of the pressure are established. Based on these results, we propose a post-processing strategy that allows us to construct an improved pressure approximation from the numerical solution. Corresponding results are well-known for mixed finite element approximations of elliptic problems and we extend these analyses here to the hyperbolic problem under consideration. We also consider the subsequent time discretization by the Crank–Nicolson method and show that the analysis and the post-processing strategy can be generalized to the fully discrete schemes. Our proofs do not rely on duality arguments or inverse inequalities and the results therefore also apply for non-convex domains and non-uniform meshes.
Mathematics Subject Classification
35L05 35L50 65L20 65M60Notes
Acknowledgements
The authors are grateful for financial support by the German Research Foundation (DFG) via Grants IRTG 1529 and TRR 154 project C4, and by the “Excellence Initiative” of the German Federal and State Governments via the Graduate School of Computational Engineering GSC 233 at Technische Universität Darmstadt.
References
- 1.Arnold, D., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Model. Math. Anal. Numer. 19, 7–32 (1985)MathSciNetCrossRefMATHGoogle Scholar
- 2.Baker, G.A.: Error estimates for finite element approximations for second order hyperbolic equations. SIAM J. Numer. Anal. 13, 564–576 (1976)MathSciNetCrossRefMATHGoogle Scholar
- 3.Baker, G.A., Bramble, J.H.: Semidiscrete and single step fully discrete approximations for second order hyperbolic equations. RAIRO Anal. Numer. 13, 75–100 (1979)MathSciNetCrossRefMATHGoogle Scholar
- 4.Bangerth, W., Rannacher, R.: Adaptive finite element techniques for the acoustic wave equation. J. Comput. Acoust. 1, 1–17 (1999)MATHGoogle Scholar
- 5.Boffi, D., Brezzi, F., Demkowicz, L.F., Durán, R.G., Falk, R.S., Fortin, M.: Mixed Finite Elements, Compatibility Conditions, and Applications, Lecture Notes in Mathematics, vol. 1939. Springer, Fondazione C.I.M.E., Berlin, Florence (2008)CrossRefGoogle Scholar
- 6.Boffi, D., Brezzi, F., Fortin, M.: Mixed finite element methods and applications. Springer Series in Computational Mathematics, vol. 44. Springer, Heidelberg (2013)CrossRefMATHGoogle Scholar
- 7.Brezzi, F., Douglas, J., Duràn, R., Fortin, M.: Mixed finite elements for second order elliptic problems in three variables. Numer. Math. 47, 237–250 (1987)MathSciNetCrossRefMATHGoogle Scholar
- 8.Brezzi, F., Douglas, J., Marini, L.D.: Two families of mixed elements for second order elliptic problems. Numer. Math. 88, 217–235 (1985)MathSciNetCrossRefMATHGoogle Scholar
- 9.Chen, Y.: Global superconvergence for a mixed finite element method for the wave equation. Syst. Sci. Math. Sci. 12, 159–165 (1999)MathSciNetMATHGoogle Scholar
- 10.Chen, Y., Huan, Y.: The superconvergence of mixed finite element methods for nonlinear hyperbolic equations. Numer. Simul. 3, 155–158 (1998)MathSciNetCrossRefGoogle Scholar
- 11.Chen, Z.: Superconvergence results for Galerkin methods for wave propagation in various porous media. Numer. Methods Partial Differ. Equ. 12, 99–122 (1999)MathSciNetCrossRefMATHGoogle Scholar
- 12.Cohen, G.: Higher-Order Numerical Methods for Transient Wave Equations. Springer, Heidelberg (2002)CrossRefMATHGoogle Scholar
- 13.Cohen, G., Joly, O., Roberts, J.E., Tordjman, N.: Higher order triangular finite elements with mass lumping for the wave equation. SIAM J. Numer. Anal. 38, 2047–2078 (2001)MathSciNetCrossRefMATHGoogle Scholar
- 14.Cowsar, L.C., Dupont, T.F., Wheeler, M.F.: A priori estimates for mixed finite element approximations of second-order hyperbolic equations with absorbing boundary conditions. SIAM J. Numer. Anal. 33, 492–504 (1996)MathSciNetCrossRefMATHGoogle Scholar
- 15.Douglas, J., Dupont, T., Wheeler, M.F.: A quasi-projection analysis of Galerkin methods for parabolic and hyperbolic equations. Math. Comput. 32, 345–362 (1978)MathSciNetCrossRefMATHGoogle Scholar
- 16.Dupont, T.: \(l^2\) estimates for Galerkin methods for second-order hyperbolic equations. SIAM J. Numer. Anal. 10, 880–889 (1973)MathSciNetCrossRefMATHGoogle Scholar
- 17.Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence (1998)Google Scholar
- 18.Geveci, T.: On the application of mixed finite element methods to the wave equations. RAIRO Model. Mathods Anal. Numer. 22, 243–250 (1988)MathSciNetCrossRefMATHGoogle Scholar
- 19.Jenkins, E.W., Rivière, T., Wheeler, M.F.: A priori error estimates for mixed finite element approximations of the acoustic wave equation. SIAM J. Numer. Anal. 40, 1698–1715 (2002)MathSciNetCrossRefMATHGoogle Scholar
- 20.Joly, P.: Variational methods for time-dependent wave propagation problems. In: Ainsworth, M., Davies, P., Duncan, D., Martin, P., Rynne, B. (eds.) Topics in Computational Wave Propagation, LNCSE, vol. 31, pp. 201–264. Springer, BerlinGoogle Scholar
- 21.Karaa, S.: Error estimates for finite element approximations of a viscous wave equation. Numer. Func. Anal. Optim. 32, 750–767 (2011)MathSciNetCrossRefMATHGoogle Scholar
- 22.Landau, L.D., Lifshitz, E.M.: Course of theoretical physics, vol. 6, 2nd edn. Pergamon Press, Oxford, 1987. Fluid mechanics, Translated from the third Russian edition by J. B. Sykes and W. H. ReidGoogle Scholar
- 23.Makridakis, C.G.: On mixed finite element methods for linear elastodynamics. Numer. Math. 61, 235–260 (1992)MathSciNetCrossRefMATHGoogle Scholar
- 24.Makridakis, C.G., Monk, P.: Time-discrete finite element schemes for Maxwell’s equations. RAIRO Model. Mathods Anal. Numer. 29, 171–197 (1995)MathSciNetCrossRefMATHGoogle Scholar
- 25.Monk, P.: Analysis of a finite element methods for Maxwell’s equations. SIAM J. Numer. Anal. 29, 714–729 (1992)MathSciNetCrossRefMATHGoogle Scholar
- 26.Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. Springer, New York (1983)CrossRefMATHGoogle Scholar
- 27.Schöberl, J.: Commuting quasi-interpolation operators for mixed finite elements. Isc-01-10-math, Institute for Scientific Computing, Texas A&M University (2001)Google Scholar
- 28.Stenberg, R.: Postprocessing schemes for some mixed finite elements. RAIRO Model. Mathods Anal. Numer. 25, 151–167 (1991)MathSciNetCrossRefMATHGoogle Scholar
- 29.Wang, F., Chen, Y., Tang, Y.: Superconvergence of fully discrete splitting positive definite mixed FEM for hyperbolic equations. Numer. Methods Partial Differ. Equ. 30, 175–186 (2014)MathSciNetCrossRefMATHGoogle Scholar
- 30.Wang, X., Bathe, K.-J.: Displacement/pressure based mixed finite element formulations for acoustic fluid–structure interaction problems. Int. J. Numer. Methods Eng. 40, 2001–2017 (1997)CrossRefMATHGoogle Scholar