Superconvergence analysis of a mixed finite element approximation for the nonlinear fourth-order Rosenau-RLW equation

Abstract

The focus of this paper is on a implicit Backward-Euler (BE) scheme with the mixed finite element method (FEM) for the two-dimentional general Rosenau-RLW equation. In which the bilinear element is used to approximate the exact solution v and the variable p=−Δv, and the zero-order nédéléc's finite element to the variable q→=ϕ∇v, respectively. An important point is that with the proposed scheme we are able to bound the numerical solution in H1-norm, so that we can efficiently solve the nonlinear term. Moreover, the stability, existence and uniqueness of approximate solution are demonstrated. Based on the combination of the interpolation and projection technique, the superconvergence estimates of order O(h2+τ) for v in H1-norm and the introducing variable q→ in L2-norm, and optimal error estimate of order O(h+τ) for introducing variable p in H1-norm is proved. Finally, numerical example is done to certify our theoretical results.