Extrapolation of Finite Element Approximation in a Rectangular Domain (Journal of Computational Mathematics, 1989)
Recently, the Richardson extrapolation for the elliptic Ritz projection with linear triangular elements on a general convex polygonal domain was discussed by Lin and Lu. We go back in this note to the simplest case, i.e. the bilinear rectangular elements on a rectangular domain which is a parallel case of the one-triangle model in the early work of Lin and Liu. We find that the finite element argument for the Richardson extrapolation with an accuracy of $O(h^4)$ needs only the regularity of $H^{4,∞}$ for the solution $u$ but the finite difference argument for extrapolation with $O(h^{s+α})$ accuracy needs $u\in C^{5+α}(0＜α＜1)$. Moreover, a formula is suggested to guarantee the extrapolation of $O(h^4)$ accuracy at fine gridpoints as well as at coarse gridpoints.