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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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##### References:
 [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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