Optimal error estimation for Petrov-Galerkin methods in two dimensions (Q1203401)

The following nonselfadjoint problem is studied for the convection- diffusion equation in two dimensions: \(-a\Delta u+\bigtriangledown\cdot (\mathbf{b}u)=f\) in \(\Omega\), \(u=g\) on \(\Gamma_ D\), \(\partial u/\partial n=0\) on \(\Gamma_ N\). Here \(\Omega\subset \mathbb{R}^ 2\) is a bounded open set with a Lipschitz continuous boundary \(\Gamma=\Gamma_ D\cup \Gamma_ N\), \(\Gamma_ D\cap \Gamma_ N=\emptyset\), and meas\((\Gamma_ D)>0\). To obtain optimal error bounds for the Petrov-Galerkin method used, the concept of symmetrization is exploited. It involves the Riesz representation theorem and a given symmetric continuous bilinear form. A decomposition of the space \(L^ 2(\Omega)^ 2\) into the direct sum of curl-free and divergence-free subspaces enables one to eliminate the Riesz operator from the approximate problem. Specific examples illustrate the general developments.