Uniform Convergence and Superconvergence of Mixed Finite Element Methods on Anisotropically Refined Grids

The lowest order Raviart--Thomas rectangular element is considered for solving the singular perturbation problem $-\mbox{div}(a\nabla p)+bp=f,$ where the diagonal tensor $a=(\varepsilon^2,1)$ or $a=(\varepsilon^2,\varepsilon^2).$ Global uniform convergence rates of O(N-1) for both p and a1/2\nabla p$ in the L2-norm are obtained in both cases, where N is the number of intervals in either direction. The pointwise interior (away from the boundary layers) convergence rates of O(N-1) for p are also proved. Superconvergence (i.e., O(N-2)) at special points and O(N-2) global L2 estimate for both p and $a^{1/2}\nabla p$ are obtained by a local postprocessing. Numerical results support our theoretical analysis. Moreover, numerical experiments show that an anisotropic mesh gives more accurate results than the standard global uniform mesh.