Superconvergence for triangular cubic elements

A new structure of superconvergence for the cubic triangular finite element approximation u/2 to a second-order elliptic problem Au = f is studied based on some orthogonal expansions in an interval. Suppose that Ω \s a convex polygonal domain with boundary Γ, its triangulation is uniform and Th is a set of vertexes and side midpoints of all elements. Then uh itself has no superconvergence points in Ω, while in any interior subdomain Ω0 the average gradient Duh has superconvergence D(dh - u) = O(hm+1 lnh) at z ∈ Th∩Ω0(no other superconvergence points). Furthermore, prescribe u = 0 on Γ1. Then the superconvergence near Γ1 will surely disappear; if α vu + bu = 0 on Γ3, where v is the conormal direction, the numercal experiments show superconvergence up to Γ3 (the case of A =-Δ and b = 0 has already been proved).