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Finite element computations for second-order hyperbolic equations with non-smooth solutions. (English) Zbl 0679.65086
This paper is concerned with approximations to non-smooth solutions of the initial boundary value problem \(u_{tt}=u_{xx}\) \(0<x<\pi\), \(0<t\leq \pi\), \(u(0,t)=u(\pi,t)=0,\) \(u(x,0)=u_ 0(x),\) \(0\leq x\leq \pi\), \(u_ t(x,0)=0\), \(0\leq x\leq \pi\). The problems of interest are those for which \(u_ 0(x)\) has a jump discontinuity. An example considered is \(u_ 0=\pi /2-| x-\pi /2|.\) The discrete approximations have been presented elsewhere and the main purpose of the paper is the generalization to discontinuous \(u_ 0(x)\). It is found that it is necessary to use a smoothing approximation to the solution of the form \(U_{new}(x)=\int \kappa (x-y)U_{old}(y)dy.\)
Reviewer: B.Burrows

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35L05 Wave equation
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[1] and , ’New Aspects of cosine schemes for second order hyperbolic equations’, in Advances in Computer Methods for Partial Differential Equations, V. R. Vichnevetsky and R. S. Stepleman (Eds.), IMACS-1984, pp. 540-545.
[2] Serbin, Appl. Math. Comput. 5 pp 57– (1979)
[3] Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1984. · Zbl 0528.65052
[4] Bramble, Math. Comput. 31 pp 94– (1977)
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