A posteriori $L^\infty(L^2)$-error bounds in finite element approximation of the wave equation

Source: arXiv

ABSTRACT We address the error control of Galerkin discretization (in space) of linear
second order hyperbolic problems. More specifically, we derive a posteriori
error bounds in the L\infty(L2)-norm for finite element methods for the linear
wave equation, under minimal regularity assumptions. The theory is developed
for both the space-discrete case, as well as for an implicit fully discrete
scheme. The derivation of these bounds relies crucially on carefully
constructed space- and time-reconstructions of the discrete numerical
solutions, in conjunction with a technique introduced by Baker (1976, SIAM J.
Numer. Anal., 13) in the context of a priori error analysis of Galerkin
discretization of the wave problem in weaker-than-energy spatial norms.

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    ABSTRACT: We study a finite element method applied to a system of coupled wave equations in a bounded smooth domain in \mathbbRd {\mathbb{R}^d} , d = 1, 2, 3, associated with a locally distributed damping function. We start with a spatially continuous finite element formulation allowing jump discontinuities in time. This approach yields, L 2(L 2) and L ∞(L 2), a posteriori error estimates in terms of weighted residuals of the system. The proof of the a posteriori error estimates is based on the strong stability estimates for the corresponding adjoint equations. Optimal convergence rates are derived upon the maximal available regularity of the exact solution and justified through numerical examples. Bibliography: 14 titles. Illustrations: 4 figures.
    Journal of Mathematical Sciences 01/2011; 175(3):228-248.

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