Superconvergence of unfitted Rannacher-Turek nonconforming element for elliptic interface problems

Abstract

The main aim of this paper is to study the superconvergence of nonconforming Rannacher-Turek finite element for elliptic interface problems under unfitted square meshes. In particular, we analyze its superclose property between the gradient of the numerical solution and the gradient of the interpolation of the exact solution. Moreover, we introduce a postprocessing interpolation operator which is applied to numerical solution, and we prove that the postprocessed gradient converges to the exact gradient with a superconvergent rate $O\left(h^{\frac{3}{2}}\right)$. Finally, numerical results coincide with our theoretical analysis, and they show that the error estimates do not depend on the ratio of the discontinuous coefficients.