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DarcyLite is a Matlab toolbox developed for numerical simulations of flow and transport in porous media in two dimensions. This paper focuses on the finite element methods and the corresponding code modules for solving the Darcy equation. Specifically, four major types of finite element solvers are presented: the continuous Galerkin (CG), the discontinuous Galerkin (DG), the weak Galerkin (WG), and the mixed finite element methods (MFEM). We further discuss the main design ideas and implementation strategies in DarcyLite. Numerical examples are included to demonstrate the usage and performance of this toolbox. [ABSTRACT FROM AUTHOR]

This paper presents quadratic finite-volume methods for elliptic and parabolic problems on quadrilateral meshes that use Barlow points (optimal stress points) for dual partitions. Introducing Barlow points into the finite-volume formulations results in better approximation properties at the cost of loss of symmetry. The novel ‘symmetrization’ technique adopted in this paper allows us to derive optimal-order error estimates in the H1- and L2-norms for elliptic problems and in the L∞ (H1)- and L∞ (L2)-norms for parabolic problems. Superconvergence of the difference between the gradients of the finite-volume solution and the interpolant can also be derived. Numerical results confirm the proved error estimates. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents an error estimate in the -norm for the discontinuous Galerkin finite element methods (DGFEM) for elliptic problems with low regularity solutions. The Raviart–Thomas interpolation operator is employed to derive the new result, which complements the mesh-dependent energy norm error estimates in Gudi (2010) . Numerical results corroborate the theoretical analysis. [Copyright &y& Elsevier]

Abstract: This paper investigates convergence of the discontinuous finite volume method (DFVM) under minimal regularity assumptions on solutions of second order elliptic boundary value problems. Conventional analysis requires the solutions to be in Sobolev spaces . Here we assume the solutions are in and employ the techniques developed in Gudi (2010) to derive error estimates in a mesh-dependent energy norm and the -norm for DFVM. The theoretical estimates are illustrated by numerical results, which include problems with corner singularity and intersecting interfaces. [Copyright &y& Elsevier]

Abstract: An adaptive discontinuous finite volume method is developed and analyzed in this paper. We prove that the adaptive procedure achieves guaranteed error reduction in a mesh-dependent energy norm and has a linear convergence rate. Numerical results are also presented to illustrate the theoretical analysis. [Copyright &y& Elsevier]