How to Recover the Convergent Rate for Richardson Extrapolation on Bounded Domains (Journal of Computational Mathematics, 2018)
We are interested in solving elliptic problems on bounded convex domains by higher order methods using the Richardson extrapolation. The theoretical basis for the application of the Richardson extrapolation is the asymptotic error expansion with a remainder of higher order. Such an expansion has been derived by the method of finite difference, where, in the neighborhood of the boundary one must reject the elementary difference analogs and adopt complex ones. This plight can be changed if we turn to the method of finite elements, where no additional boundary approximation is needed but an easy triangulation is chosen, i.e. the higher order boundary approximation is replaced by a chosen triangulation. Specifically, a global error expansion with a remainder of fourth order can be derived by the linear finite element discretization over a chosen triangulation, which is obtained by decomposing the domain first and then subdividing each subdomain almost uniformly. A fourth order method can thus be constructed by the simplest linear finite element approximation over the chosen triangulation using the Richardson extrapolation.