Computer Methods in Applied Mechanics and Engineering - Special edition on the 20th Anniversary
Analysis of the efficiency of an a posteriori error estimator for linear triangular finite elements
SIAM Journal on Numerical Analysis
Wavelet methods for PDEs — some recent developments
Journal of Computational and Applied Mathematics - Special issue on numerical analysis 2000 Vol. VII: partial differential equations
Discontinuous hp-Finite Element Methods for Advection-Diffusion-Reaction Problems
SIAM Journal on Numerical Analysis
Stable Discretizations of Convection-Diffusion Problems via Computable Negative-Order Inner Products
SIAM Journal on Numerical Analysis
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We present a new approach to the a posteriori error analysis of stable Galerkin approximations of reaction-convection-diffusion problems. It relies upon a non-standard variational formulation of the exact problem, based on the anisotropic wavelet decomposition of the equation residual into convection-dominated scales and diffusion-dominated scales. The associated norm, which is stronger than the standard energy norm, provides a robust (i.e., uniform in the convection limit) control over the streamline derivative of the solution. We propose an upper estimator and a lower estimator of the error, in this norm, between the exact solution and any finite dimensional approximation of it. We investigate the behaviour of such estimators, both theoretically and through numerical experiments. As an output of our analysis, we find that the lower estimator is quantitatively accurate and robust.