Superconvergence of rectangular finite element with interpolated coefficients for semilinear elliptic problem

In this paper, the rectangular finite element method with interpolated coefficients for the second-order semilinear elliptic problem is introduced and analyzed. Assume that Ω is polygonal domain, and rectangular partition is quasiuniform and denote by Z0 and Z1 the sets of (n + 1)-order Lobatto points and n-order Gauss points of all elements, respectively. Based on an orthogonal expansion in an element, and on the property of corresponding finite element for an auxiliary linear elliptic problem, optimal superconvergence u − uh = O(hn+2), n ⩾ 2, at z ∈ Z0 and D(u − uh) = O(hn+1), n ⩾ 1, at z ∈ Z1 are proved, respectively. It is shown that the finite element with interpolated coefficients has the same superconvergence as that of classical finite elements, but more economic and efficient. Finally the results in the case of quadratic finite element are supported by a numerical example.