The Theory and Application of Upwind Finite Difference Fractional Steps Procedure for Seawater Intrusion

Numerical simulation and theoretical analysis of seawater intrusion is the mathematical basis for modern environmental science. Its mathematical model is the nonlinear coupled system of partial differential equations with initial-boundary problems. For a generic case of a three-dimensional bounded region, two kinds of finite difference fractional steps pro- cedures are put forward. Optimal order estimates in norm are derived for the error in the approximation solution. The present method has been successfully used in predicting the consequences of seawater intrusion and protection projects.

The MFE-CFE-GFE Method for the Fully Coupled Quasi-Static Thermo-Poroelastic Problem

In this work,we consider a combined finite element method for fully coupled nonlinear thermo-poroelastic model problems.The mixed finite element(MFE)method is used for the pressure,the characteristics finite element(CFE)method is used for the temperature,and the Galerkin finite element(GFE)method is used for the elastic displacement.The semi-discrete and fully discrete finite element schemes are established and the stability of this method is presented.We derive error estimates for the pressure,temperature and displacement.Several numerical examples are presented to confirm the accuracy of the method.

EXTRAPOLATION METHODS FOR COMPUTING HADAMARD FINITE-PART INTEGRAL ON FINITE INTERVALS

In this paper, we presen t the composite rectangle rule for the comp ut at ion of Hadamard finite-part integrals in boundary element methods with the hypersingular kernel l/(x-s)2 and we obtain the asymptotic expansion of error function of the middle rectangle rule. Based on the asymptotic expansion, two extrapolation algorithms are presented and their convergence rates are proved, which are the same as the Euler-Maclaurin expansions of classical middle rec tangle rule approximations. At last, some numerical results are also illustrated to confirm the theoretical results and show the efficiency of the algorithms.

LEAST-SQUARES GALERKIN PROCEDURE FOR SECOND-ORDER HYPERBOLIC EQUATIONS

This paper proposes the least-squares Galerkin finite dement scheme to solve secona-oraer hyperbolic equations. The convergence analysis shows that the method yields the approximate solutions with optimal accuracy in （L2 （Ω））2 × L2 （Ω） norms. Moreover, the method gets the approximate solutions with second-order accuracy in time increment. A numerical example testifies the efficiency of the novel scheme. Key words Convergence analysis, Galerkin finite element, hyperbolic equations, least-squares, nu- merical example.

Finite Element Approximation of Semilinear Parabolic Optimal Control Problems

In this paper,the finite element approximation of a class of semilinear parabolic optimal control problems with pointwise control constraint is studied.We discretize the state and co-state variables by piecewise linear continuous functions,and the control variable is approximated by piecewise constant functions or piecewise linear discontinuous functions.Some a priori error estimates are derived for both the control and state approximations.The convergence orders are also obtained.

A Family of Characteristic Discontinuous Galerkin Methods for Transient Advection-Diffusion Equations and Their Optimal-Order L2 Error Estimates

We develop a family of characteristic discontinuous Galerkin methods for transient advection-diffusion equations,including the characteristic NIPG,OBB,IIPG,and SIPG schemes.The derived schemes possess combined advantages of EulerianLagrangian methods and discontinuous Galerkin methods.An optimal-order error estimate in the L2 norm and a superconvergence estimate in a weighted energy norm are proved for the characteristic NIPG,IIPG,and SIPG scheme.Numerical experiments are presented to confirm the optimal-order spatial and temporal convergence rates of these schemes as proved in the theorems and to show that these schemes compare favorably to the standard NIPG,OBB,IIPG,and SIPG schemes in the context of advection-diffusion equations.

A CHARACTERISTIC FINITE ELEMENT METHOD FOR CONSTRAINED CONVECTION-DIFFUSION-REACTION OPTIMAL CONTROL PROBLEMS

In this paper, we develop a priori error estimates for the solution of constrained convection-diffusion-reaction optimal control problems using a characteristic finite element method. The cost functional of the optimal control problems consists of three parts： The first part is about integration of the state over the whole time interval, the second part refers to final-time state, and the third part is a regularization term about the control. We discretize the state and co-state by piecewise linear continuous functions, while the control is approximated by piecewise constant functions. Pointwise inequality function constraints on the control are considered, and optimal a L2-norm priori error estimates are obtained. Finally, we give two numerical examples to validate the theoretical analysis.

BLOCK-CENTERED FINITE DIFFERENCE METHODS FOR NON-FICKIAN FLOW IN POROUS MEDIA

In this article, two block-centered finite difference schemes are introduced and analyzed to solve the parabolic integro-differential equation arising in modeling non-Fickian flow in porous media. One scheme is Euler backward scheme with first order accuracy in time increment while the other is Crank-Nicolson scheme with second order accuracy in time increment. Stability analysis and second-order error estimates in spatial meshsize for both pressure and velocity in discrete L^2 norms are established on non-uniform rectangular grid. Numerical experiments using the schemes show that the convergence rates are in agreement with the theoretical analysis.

A posteriori error estimates for optimal control problems constrained by convection-diffusion equations

We propose a characteristic finite element evolutionary type convection-diffusion optimal control discretization of problems. Non- divergence-free velocity fields and bilateral inequality control constraints are handled. Then some residual type a posteriori error estimates are analyzed for the approximations of the control, the state, and the adjoint state. Based on the derived error estimators, we use them as error indicators in developing efficient multi-set adaptive meshes characteristic finite element algorithm for such optimal control problems. Finally, one numerical example is given to check the feasibility and validity of multi-set adaptive meshes refinements.

A PRIORI ERROR ESTIMATES FOR LEAST-SQUARES MIXED FINITE ELEMENT APPROXIMATION OF ELLIPTIC OPTIMAL CONTROL PROBLEMS

In this paper, a constrained distributed optimal control problem governed by a first- order elliptic system is considered. Least-squares mixed finite element methods, which are not subject to the Ladyzhenkaya-Babuska-Brezzi consistency condition, are used for solving the elliptic system with two unknown state variables. By adopting the Lagrange multiplier approach, continuous and discrete optimality systems including a primal state equation, an adjoint state equation, and a variational inequality for the optimal control are derived, respectively. Both the discrete state equation and discrete adjoint state equation yield a symmetric and positive definite linear algebraic system. Thus, the popular solvers such as preconditioned conjugate gradient （PCG） and algebraic multi-grid （AMG） can be used for rapid solution. Optimal a priori error estimates are obtained, respectively, for the control function in L2 （Ω）-norm, for the original state and adjoint state in H1 （Ω）-norm, and for the flux state and adjoint flux state in H（div; Ω）-norm. Finally, we use one numerical example to validate the theoretical findings.

A DISCONTINUOUS GALERKIN METHOD COMBINED WITH MIXED FINITE ELEMENT FOR SEAWATER INTRUSION PROBLEM

Seawater intrusion problem is considered in this paper.Its mathematical model is anonlinear coupled system of partial differential equations with initial boundary problem.It consistsof the water head equation and the salt concentration equation.A combined method is developedto approximate the water head equation by mixed finite element method and concentration equationby discontinuous Galerkin method.The scheme is continuous in time and optimal order estimates inH^1-norm and L^2-norm are derived for the errors.

Numerical Analysis of Two-Grid Block-Centered Finite Difference Method for Two-Phase Flow in Porous Medium

In this paper,a two-grid block-centered finite difference method for the incompressible miscible displacement in porous medium is introduced and analyzed,which is to solve a nonlinear equation on coarse mesh space of size H and a linear equation on fine grid of size h.We establish the full discrete two-grid block-centered finite difference scheme on a uniform grid.The error estimates for the pressure,Darcy velocity,concentration variables are derived,which show that the discrete L2 error is O(Dt+h2+H4).Finally,two numerical examples are provided to demonstrate the effectiveness and accuracy of our algorithm.

New Energy Analysis of Yee Scheme for Metamaterial Maxwell’s Equations on Non-Uniform Rectangular Meshes

In this paper,several new energy identities of metamaterial Maxwell’s equations with the perfectly electric conducting(PEC)boundary condition are proposed and proved.These new energy identities are different from the Poynting theorem.By using these new energy identities,it is proved that the Yee scheme on non-uniform rectangular meshes is stable in the discrete L2 and H1 norms when the Courant-Friedrichs-Lewy(CFL)condition is satisfied.Numerical experiments in twodimension(2D)and 3D are carried out and confirm our analysis,and the superconvergence in the discrete H1 norm is found.