The effect of spatial quadrature on finite element galerkin approximations to hyperbolic integro-differential equations |
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Authors: | R. K. Sinha A. K. Pani |
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Affiliation: | Department of Mathematics , Indian Institute of Technology , Bombay, Powai, Mumbai, 400 076, India |
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Abstract: | The purpose of this paper is to study the effect of numerical quadrature on the finite element approximations to the solutions of hyperbolic intego-differential equations. Both semidiscrete and fully discrete schemes are analyzed and optimal estimates are derived in L ∞(H 1)L ∞(L 2) norms and quasi-optimal estimate in L ∞(L ∞) norm using energy arguments. Further, optimal L(L 2)-estimates are shown to hold with minimal smoothness assumptions on the initial functions. The analysis in the present paper not only improves upon the earlier results of Baker and Dougalis [SIAM J. Numer. Anal. 13 (1976), pp. 577-598] but also confirms the minimum smoothness assumptions of Rauch [SIAM J. Numer. Anal. 22 (1985), pp. 245-249] for purely second order hyperbolic equation with quadrature. |
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