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 ?div(arp) + bp = f; where the diagonal tensor a = (" 2 ; 1) or a = (" 2 ; " 2): Global uniform convergence rates of O(N ?1) for both p and a 1=2 rp in the L 2-norm are obtained in both cases, where N is the number of intervals in both directions. 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 L 2 estimate for both p and a 1=2 5 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.