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Semidiscrete and single step fully discrete finite element approximations for second order hyperbolic equations with nonsmooth solutions. (English) Zbl 0766.65082
The author considers finite element approximations for an initial- boundary value problem attached to a second order hyperbolic equation. The spatial operator is a second order uniformly elliptic one.
For semidiscrete approximations and for fully discrete approximations (based on rational approximations of \(\exp(-z))\) convergence estimates in negative norms are obtained. \(L^ 2\) projections of the initial data as starting values are used in both cases.

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M20 Method of lines 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
35L15 Initial value problems for second-order hyperbolic equations
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References:
[1] G. A. BAKER and J. H. BRAMBLESemidiscrete and single step fully discrete approximations for second order hyperbolic equations, RAIRO Modél. Math. Anal. Numer., V. 13, 1979, pp. 75-100. Zbl0405.65057 MR533876 · Zbl 0405.65057 · eudml:193340
[2] L. A. BALES, Finite element computations for second order hyperbolic equations with nonsmooth solutions, Comm. in App. Num. Meth., V. 5, 1989, pp. 383-388. Zbl0679.65086 · Zbl 0679.65086 · doi:10.1002/cnm.1630050604
[3] J. H. BRAMBLE and A. H. SCHATZ, Higher order local accuracy by averaging in the finite element method, Math. Comp., V. 31, 1977, pp. 94-111. Zbl0353.65064 MR431744 · Zbl 0353.65064 · doi:10.2307/2005782
[4] T. GEVECI, On the convergence of Galerkin approximation schemas for second-order hyperbolic equations in energy and negative norms, Math. Comp., V. 42, 1984, pp.393-415. Zbl0553.65082 MR736443 · Zbl 0553.65082 · doi:10.2307/2007592
[5] C. JOHNSON and U. NAVERT, An analysis of some finite element methods for advection-diffusion problems, in Analytical and Numerical Approaches to Asymptotic Problems in Analysis, L. S. Frank and A. van der Sluis (Eds), North-Holland, 1981, pp. 99-116. Zbl0455.76081 MR605502 · Zbl 0455.76081
[6] P. D. LAX and M. S. MOCK, The computation of discontinuous solutions of linear hyperbolic equations, Comm Pure Appl. Math., V. 31, 1978, pp. 423-430. Zbl0362.65075 MR468216 · Zbl 0362.65075 · doi:10.1002/cpa.3160310403
[7] V. THOMEE, Galerkin Finite Methods for Parabolic Problems, Springer-Verlag, 1984. Zbl0528.65052 MR744045 · Zbl 0528.65052 · doi:10.1007/BFb0071790
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