Mixed Discontinuous Finite Element Research Articles

A mixed discontinuous finite element method for folded Naghdi’s shell in Cartesian coordinates

A mixed discontinuous finite element method for folded Naghdi’s shell in Cartesian coordinates

Mixed Discontinuous Finite Element Methods for Multiphase Flow in Porous Media

In this paper we present an application of mixed discontinuous finite element methods to the simulation of two phase immiscible flow in porous media. The partial differential system describing this flow is written in terms of an elliptic equation for a global pressure and a parabolic equation for a saturation. Both of these two equations are solved using the mixed discontinuous finite element methods. Numerical results are presented for P0, P1, and P2 discontinuous elements. The P0 results are very close to thoseb generated by the standard finite difference method, while the P1 and P2 results seem more accurate.

Preconditioning a mixed discontinuous finite element method for radiation diffusion

AbstractWe propose a multilevel preconditioning strategy for the iterative solution of large sparse linear systems arising from a finite element discretization of the radiation diffusion equations. In particular, these equations are solved using a mixed finite element scheme in order to make the discretization discontinuous, which is imposed by the application in which the diffusion equation will be embedded. The essence of the preconditioner is to use a continuous finite element discretization of the original, elliptic diffusion equation for preconditioning the discontinuous equations. We have found that this preconditioner is very effective and makes the iterative solution of the discontinuous diffusion equations practical for large problems. This approach should be applicable to discontinuous discretizations of other elliptic equations. We show how our preconditioner is developed and applied to radiation diffusion problems on unstructured, tetrahedral meshes and show numerical results that illustrate its effectiveness. Published in 2004 by John Wiley & Sons, Ltd.

Stability and convergence of mixed discontinuous finite element methods for second-order differential problems

In this paper we develop an abstract theory for stability and convergence of mixed discontinuous finite element methods for second-order partial differential problems. This theory is then applied to various examples, with an emphasis on different combinations of mixed finite element spaces. Elliptic, parabolic, and convection-dominated diffusion problems are considered. The examples include classical mixed finite element methods in the discontinuous setting, local discontinuous Galerkin methods, and their penalized (stablized) versions. For the convection-dominated diffusion problems, a characteristics-based approach is combined with the mixed discontinuous methods.

Numerical study of the HP version of mixed discontinuous finite element methods for reaction‐diffusion problems: The 1D case

AbstractThis article studies the stability and convergence of the hp version of the three families of mixed discontinuous finite element (MDFE) methods for the numerical solution of reaction‐diffusion problems. The focus of this article is on these problems for one space dimension. Error estimates are obtained explicitly in the grid size h, the polynomial degree p, and the solution regularity; arbitrary space grids and polynomial degree are allowed. These estimates are asymptotically optimal in both h and p for some of these methods. Extensive numerical results to show convergence rates in h and p of the MDFE methods are presented. Theoretical and numerical comparisons between the three families of MDFE methods are described. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 525–553, 2003

The hp version of Eulerian‐Lagrangian mixed discontinuous finite element methods for advection‐diffusion problems

We study the hp version of three families of Eulerian‐Lagrangian mixed discontinuous finite element (MDFE) methods for the numerical solution of advection‐diffusion problems. These methods are based on a space‐time mixed formulation of the advection‐diffusion problems. In space, they use discontinuous finite elements, and in time they approximately follow the Lagrangian flow paths (i.e., the hyperbolic part of the problems). Boundary conditions are incorporated in a natural and mass conservative manner. In fact, these methods are locally conservative. The analysis of this paper focuses on advection‐diffusion problems in one space dimension. Error estimates are explicitly obtained in the grid size h, the polynomial degree p, and the solution regularity; arbitrary space grids and polynomial degree are allowed. These estimates are asymptotically optimal in both h and p for some of these methods. Numerical results to show convergence rates in h and p of the Eulerian‐Lagrangian MDFE methods are presented. They are in a good agreement with the theory.

Stability and convergence of mixed discontinuous finite element methods for second-order differential problems

In this paper we develop an abstract theory for stability and convergence of mixed discontinuous finite element methods for second-order partial differential problems. This theory is then applied to various examples, with an emphasis on different combinations of mixed finite element spaces. Elliptic, parabolic, and convection-dominated diffusion problems are considered. The examples include classical mixed finite element methods in the discontinuous setting, local discontinuous Galerkin methods, and their penalized (stablized) versions. For the convection-dominated diffusion problems, a characteristics-based approach is combined with the mixed discontinuous methods.

Effects of boundary conditions on a class of mixed discontinuous finite element methods for reaction–diffusion problems

AbstractThis paper numerically studies the effects of boundary conditions of different types on a class of mixed discontinuous finite element methods. The focus of this study is on a reaction–diffusion problem in one space dimension, which arises in the modelling and simulation of fluid flows in porous media. Numerical results show that for boundary conditions of certain types these mixed discontinuous methods produce an optimal rate of convergence; for others they generate a sub‐optimal rate of convergence. A lower‐order term is included in the reaction–diffusion problem, so the numerical study applies to a time‐dependent problem as well. Copyright © 2002 John Wiley & Sons, Ltd.

Characteristic mixed discontinuous finite element methods for advection-dominated diffusion problems

In this paper three characteristic mixed discontinuous finite element methods are introduced for time dependent advection-dominated diffusion problems. Namely, the diffusion term in these problems is discretized using mixed discontinuous finite elements, and the temporal differentiation and advection terms are treated by a characteristic tracking scheme. The first method is based on the standard modified method of characteristics. It is simple to set up and analyze, but fails to preserve an integral identity satisfied by the solution of the differential problems. The second method is formulated using the modified method of characteristics with adjusted advection and preserves the integral identity globally. The third method is defined in terms of a local Eulerian–Lagrangian technique and preserves the identity locally. These three methods not only preserve the conceptual and computational merits of both characteristics-based procedures and discontinuous finite element schemes, they also possess new features such as they are more stable, are more accurate, and can handle the case where the diffusion coefficient is zero. Stability and convergence properties are studied for all three methods; unconditionally stable results and sharp error estimates are established. Their relationships to standard characteristics-based methods such as MMOC, MMOCAA, ELLAM, and CMFEM are described in detail. Numerical results are presented.