UC San Diego

In this dissertation, we develop new superconvergence estimates of mixed and nonconforming finite element methods on mildly structured grids, where most pairs of adjacent triangles form approximate parallelograms. In particular, we consider the Raviart--Thomas mixed method and Crouzeix--Raviart nonconforming method for second order elliptic equations, and the Hellan--Herrmann--Johnson mixed method and Morley nonconforming method for fourth order elliptic equations. We first prove some supercloseness estimates, that is, the canonical interpolant and finite element solution are superclose. We then develop a new family of recovery operators on irregular triangular grids by using the idea of local least-squares fittings. Combining these two ingredients, we prove that the postprocessed solution superconverges to the exact solution. Compared to existing superconvergence results, our estimates are sharper and applicable to more flexible grids and partial differential equations with variable coefficients and lower order terms.