Showing papers by "Mary F. Wheeler published in 2000"

Mixed Finite Element Methods on Nonmatching Multiblock Grids

Abstract: We consider mixed finite element methods for second order elliptic equations on nonmatching multiblock grids. A mortar finite element space is introduced on the nonmatching interfaces. We approximate in this mortar space the trace of the solution, and we impose weakly a continuity of flux condition. A standard mixed finite element method is used within the blocks. Optimal order convergence is shown for both the solution and its flux. Moreover, at certain discrete points, superconvergence is obtained for the solution and also for the flux in special cases. Computational results using an efficient parallel domain decomposition algorithm are presented in confirmation of the theory.

A Discontinuous Galerkin Method Applied to Nonlinear Parabolic Equations

Abstract: Semi-discrete and a family of discrete time locally conservative Discontinuous Galerkin procedures are formulated for approximations to nonlinear parabolic equations. For the continuous time approximations a priori L ∞ (L 2) and L 2(H l) estimates are derived and similarly, l ∞ (L 2) and l 2 (H 1) for the discrete time schemes. Spatial rates in H l and time truncation errors in L 2 are optimal.

Part II. Discontinuous Galerkin method applied to a single phase flow in porous media

Abstract: Discontinuous Galerkin numerical simulations of single phase flow problem are described in this paper. The simulations show the advantages of using discontinuous approximation spaces. hp convergence results are obtained for smooth solutions. Unstructured meshes and unsmooth solutions are also considered.

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

Abstract: The lowest order Raviart--Thomas rectangular element is considered for solving the singular perturbation problem $-\mbox{div}(a abla p)+bp=f,$ 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 abla 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 $a^{1/2} abla p$ 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.