Positive Definite Mixed Element Method Research Articles

Parallel characteristic mixed element method for saltwater intrusion problem

Numerical simulation of saltwater intrusion problem has attracted more and more attention in the past decades. Based on parallel subspace correction method proposed by Xu (1992), a new overlapping domain decomposition parallel algorithm is constructed to solve saltwater intrusion problem. In this algorithm, the splitting positive definite mixed element method is used to discrete head water equation, while the characteristic Galerkin finite element method is used to approximate concentration equation. The relation between the convergence rate and discretization parameters, including the overlapping degree of the subspaces, is considered. Furthermore, the convergence of this algorithm is analyzed and the corresponding error estimate is given. The convergence result suggests that only two iterations are needed to reach the given accuracy at each time level.

The splitting mixed element method for parabolic equation and its application in chemotaxis model

In this article, we first revisit the splitting positive definite mixed element method for reaction-diffusion equation, in which the mixed system is symmetric positive definite. And then we apply this technique to the variable coefficient parabolic equation and give the corresponding fully-discrete scheme with second-order central difference formula in time. We study the convergence of the semi-discrete and fully-discrete scheme and derive the error estimates. Finally, we extend this method to chemotaxis model and give the corresponding numerical results, which suggests that it has the ability to recover blowing-up solutions.

A Combined Discontinuous Galerkin Method for Saltwater Intrusion Problem with Splitting Mixed Procedure

AbstractIn this paper, a new combined method is presented to simulate saltwater intrusion problem. A splitting positive definite mixed element method is used to solve the water head equation, and a symmetric discontinuous Galerkin (DG) finite element method is used to solve the concentration equation. The introduction of these two numerical methods not only makes the coefficient matrixes symmetric positive definite, but also does well with the discontinuous problem. The convergence of this method is considered and the optimal L2-norm error estimate is also derived.

Splitting positive definite mixed element method for viscoelasticity wave equation

A splitting positive definite mixed finite element method is proposed for second-order viscoelasticity wave equation. The proposed procedure can be split into three independent symmetric positive definite integro-differential sub-system and does not need to solve a coupled system of equations. Error estimates are derived for both semidiscrete and fully discrete schemes. The existence and uniqueness for semidiscrete scheme are proved. Finally, a numerical example is provided to illustrate the efficiency of the method.

Splitting positive definite mixed element methods for pseudo‐hyperbolic equations

AbstractSplitting positive definite mixed finite element (SPDMFE) methods are discussed for a class of second‐order pseudo‐hyperbolic equations. Depending on the physical quantities of interest, two methods are proposed. Error estimates are derived for both semidiscrete and fully discrete schemes. The existence and uniqueness for semidiscrete schemes are proved. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 670–688, 2012

A splitting positive definite mixed element method for second‐order hyperbolic equations

AbstractIn this article, we establish a new mixed finite element procedure, in which the mixed element system is symmetric positive definite, to solve the second‐order hyperbolic equations. The convergence of the mixed element methods with continuous‐ and discrete‐time scheme is proved. And the corresponding error estimates are given. Finally some numerical results are presented. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009

A splitting positive definite mixed element method for miscible displacement of compressible flow in porous media

AbstractA miscible displacement of one compressible fluid by another in a porous medium is governed by a nonlinear parabolic system. A new mixed finite element method, in which the mixed element system is symmetric positive definite and the flux equation is separated from pressure equation, is introduced to solve the pressure equation of parabolic type, and a standard Galerkin method is used to treat the convection‐diffusion equation of concentration of one of the fluids. The convergence of the approximate solution with an optimal accuracy in L2‐norm is proved. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 229–249, 2001