Extrapolation of numerical solutions for elliptic problems on corner domains (Q1354157)

The paper is devoted to the study and development of extrapolation techniques for the numerical solution of elliptic boundary value problems on corner domains. As a rule, the application of extrapolation techniques for boundary value problems on corner domains is difficult because the requirements of smoothness assumptions on the exact solution do not hold for these problems. The authors solve this deficiency for the model problem \[ -\Delta u=f, \quad\text{in } \Omega, \qquad u=g, \quad\text{on } \partial\Omega \tag{1} \] and state that extrapolation techniques can be successfully applied without extra smoothness assumption on the exact solution. Furthermore, in asymptotic error expansions, higher order terms exist, depending on the smoothness of the given data other than the domain. So, the main theoretical result of the paper is an asymptotic error expansion of the numerical solution of the Poisson equation (1) on rectangles. The smoothness only of \(f\) and \(g\) is assumed without any smoothness assumptions on the exact solution. The result is proved only for the model problem (1) on a rectangle. However, one might expect on the form of asymptotic error expansions and when the asymptotic error expansions might be true, for numerical solutions of more general problems with more general meshes. Some numerical results are also presented.