Article

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

03/2010;
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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