Summary
We examine the optimality of conforming Petrov-Galerkin approximations for the linear convection-diffusion equation in two dimensions. Our analysis is based on the Riesz representation theorem and it provides an optimal error estimate involving the smallest possible constantC. It also identifies an optimal test space, for any choice of consistent norm, as that whose image under the Riesz representation operator is the trial space. By using the Helmholtz decomposition of the Hilbert space [L 2(Ω)]2, we produce a construction for the constantC in which the Riesz representation operator is not required explicitly. We apply the technique to the analysis of the Galerkin approximation and of an upwind finite element method.
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Morton, K.W., Murdoch, T. & Süli, E. Optimal error estimation for Petrov-Galerkin methods in two dimensions. Numer. Math. 61, 359–372 (1992). https://doi.org/10.1007/BF01385514
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DOI: https://doi.org/10.1007/BF01385514