Ultraconvergence of high order FEMs for elliptic problems with variable coefficients

Abstract

In this paper, we investigate local ultraconvergence properties of the high-order finite element method (FEM) for second order elliptic problems with variable coefficients. Under suitable regularity and mesh conditions, we show that at an interior vertex, which is away from the boundary with a fixed distance, the gradient of the post-precessed kth $$(k\ge 2)$$(kź2) order finite element solution converges to the gradient of the exact solution with order $$\mathcal{O}(h^{k+2} (\mathrm{ln} h)^3)$$O(hk+2(lnh)3). The proof of this ultraconvergence property depends on a new interpolating operator, some new estimates for the discrete Green's function, a symmetry theory derived in [26], and the Richardson extrapolation technique in [20]. Numerical experiments are performed to demonstrate our theoretical findings.