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On the application of mixed finite element methods to the wave equations. (English) Zbl 0646.65083
This paper examines the convergence of some semi-discrete approximations to the wave equation in a general space which includes the so-called Raviart-Thomas space as a special case. In addition, a brief discussion of the solution of the ensuing system of ordinary differential equations using the implicit Euler method is presented.
Reviewer: K.Burrage

65N40 Method of lines for boundary value problems involving PDEs
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
35L05 Wave equation
Full Text: DOI EuDML
[1] R. ALEX ANDER, Diagonally implicity Runge-Kutta methods for stiff O.D.E.’s, SIAM J. Numer. Anal. 14 (1977), 1006-1021. Zbl0374.65038 MR458890 · Zbl 0374.65038
[2] [2] D. N. ARNOLD and F. BREZZI, Mixed and non conforming finite elementmethods: Implementation, postprocessing and error estimates, R.A.R.O. Math. Model, and Num. Anal. (M2AN) 1 (1985), 7-32. Zbl0567.65078 MR813687 · Zbl 0567.65078
[3] [3] D. N. ARNOLD, J. DOUGLAS, Jr., and C. P. GUPTA, A family of higher ordermixed finite element methods for plane elasticity, Numer. Math. 45 (1984), 1-22. Zbl0558.73066 MR761879 · Zbl 0558.73066
[4] [4] G. A. BAKER and J. H. BRAMBLE, Semidiscrete and single step fully discreteapproximations for second order hyperbolic equations, R.A.I.R.O. Anal. Num. 13 (1979), 75-100. Zbl0405.65057 MR533876 · Zbl 0405.65057
[5] [5] B. BRENNER, M. CROUZEIX and V. THOMÉE, Single step methods for inhomogeneous linear differential equations in Banach spaces, R.A.I.R.O. Anal. Num. 16 (1982), 5-26. Zbl0477.65040 MR648742 · Zbl 0477.65040
[6] [6] F. BREZZI, J. DOUGLAS Jr. and L. D MARINI, Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985), 217-235. Zbl0599.65072 MR799685 · Zbl 0599.65072
[7] K. BURRAGE, Efficiently implementable algebraically stable Runge-Kutta methods, SIAM J. Numer. Anal. 19 (1982), 245-258. Zbl0483.65040 MR650049 · Zbl 0483.65040
[8] M. CROUZEIX, Sur l’approximation des équations différentielles opérationnelles linéaires par des méthodes de Runge-Kutta, thèse, Paris (1975).
[9] M. CROUZEIX and P.-A. RAVIART, Approximation des problèmes d’évolution, preprint, Université de Rennes (1980).
[10] V. DOUGALIS and S.M. SERBIN, One some unconditionally stable, higher order methods for numerical solution of the structural dynamics equations, Int. J. Num. Meth. Eng. 18 (1982), 1613-1621. Zbl0488.73087 MR680513 · Zbl 0488.73087
[11] E. GEKELER, Discretization Methods for Stable Initial Value Problems, Springer Lecture Notes in Mathematics 1044 (1984), Springer-Verlag, Berlin, Heidelberg, New York. Zbl0518.65050 MR731695 · Zbl 0518.65050
[12] [12] C. JOHNSON and V. THOMÉE, Error estimates for some mixed finite element methodes for parabolic type problems, R.A.I.R.O. Anal. Num. 15 (1981), 41-78. Zbl0476.65074 MR610597 · Zbl 0476.65074
[13] P.-A., RAVIART and J.M. THOMAS, A mixed finite element method for 2nd order problems, in Mathematical Aspects of the Element Method, Springer Lecture Notes in Mathematics 606 (1977), Springer-Verlag, Berlin-Heidelberg-New York. MR483555 · Zbl 0362.65089
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