Full-Discrete Weak Galerkin Finite Element Method for Solving Diffusion-Convection Problem.

This paper applied and analyzes full discrete weak Galerkin (WG) finite element method for non steady two dimensional convection-diffusion problem on conforming polygon. We approximate the time derivative by backward finite difference method and the elliptic form by WG finite element method. The main idea of WG finite element methods is the use of weak functions and their corresponding discrete weak derivatives in standard weak form of the model problem. The theoretical evidence proved that the error estimate in  norm, the properties of the bilinear form, (v-elliptic and continuity), stability, and the energy conservation law

Mixed and discontinuous finite volume element schemes for the optimal control of immiscible flow in porous media

We introduce a family of hybrid discretisations for the numerical approximation of optimal control problems governed by the equations of immiscible displacement in porous media. The proposed schemes are based on mixed and discontinuous finite volume element methods in combination with the optimise-then-discretise approach for the approximation of the optimal control problem, leading to nonsymmetric algebraic systems, and employing minimum regularity requirements. Estimates for the error (between a local reference solution of the infinite dimensional optimal control problem and its hybrid approximation) measured in suitable norms are derived, showing optimal orders of convergence

Use of discontinuity factors in high-order finite element methods

The discontinuity factors are a technique widely used in nodal methods to minimize the error due to spatial homogenization of cross sections for a coarse mesh core calculation. In the present work, the introduction of discontinuity factors in a high-order finite element approximation of the neutron diffusion equation is investigated. More precisely, classical reference and assembly discontinuity factors are introduced in a discontinuous Galerkin finite element method stabilized using an interior penalty formulation for the neutron diffusion equation. The proposed method is tested solving different one- and two-dimensional benchmark problems, showing that the discontinuity factors technique can be successfully introduced in the discontinuous Galerkin formulation. (C) 2015 Published by Elsevier Ltd.The work performed by the second, fifth, and sixth author was financially supported by the Swedish Research Council (VR - Vetenskapsradet) within a framework grant called DREAM4SAFER (Development of Revolutionary and Accurate Methods for Safety Analyses of Future and Existing Reactors), research contract C0467701.Vidal-Ferràndiz, A.; González Pintor, S.; Ginestar Peiro, D.; Verdú Martín, GJ.; Asadzadeh, M.; Demazière, C. (2016). Use of discontinuity factors in high-order finite element methods. Annals of Nuclear Energy. 87:728-738. https://doi.org/10.1016/j.anucene.2015.06.021S7287388

Postprocessing of Non-Conservative Flux for Compatibility with Transport in Heterogeneous Media

A conservative flux postprocessing algorithm is presented for both steady-state and dynamic flow models. The postprocessed flux is shown to have the same convergence order as the original flux. An arbitrary flux approximation is projected into a conservative subspace by adding a piecewise constant correction that is minimized in a weighted $L^2$ norm. The application of a weighted norm appears to yield better results for heterogeneous media than the standard $L^2$ norm which has been considered in earlier works. We also study the effect of different flux calculations on the domain boundary. In particular we consider the continuous Galerkin finite element method for solving Darcy flow and couple it with a discontinuous Galerkin finite element method for an advective transport problem.Comment: 34 pages, 17 figures, 11 table

INTERIOR PENALTY DISCONTINUOUS GALERKIN METHODS WITH IMPLICIT TIME-INTEGRATION TECHNIQUES FOR NONLINEAR PARABOLIC EQUATIONS

Abstract. We prove existence and numerical stability of numerical solutions of three fully discrete interior penalty discontinuous Galerkin (IPDG) methods for solving nonlinear parabolic equations. Under some appropriate regularity conditions, we give the l 2 (H 1 ) and l ∞ (L 2 ) error estimates of the fully discrete symmetric interior penalty discontinuous Galerkin (SIPG) scheme with the implicit θ-schemes in time, which include backward Euler and Crank-Nicolson finite difference approximations. Our estimates are optimal with respect to the mesh size h. The theoretical results are confirmed by some numerical experiments