An expanded mixed covolume method for elliptic problems

We consider the mixed covolume method combining with the expanded mixed element for a system of first‐order partial differential equations resulting from the mixed formulation of a general self‐adjoint elliptic problem with a full diffusion tensor. The system can be used to model the transport of a contaminant carried by a flow in porous media. We use the lowest order Raviart‐Thomas mixed element space. We show the first‐order error estimate for the approximate solution in L² norm. We show the superconvergence both for pressure and velocity in certain discrete norms. We also get a finite difference scheme by using proper approximate integration formulas. Finally we give some numerical examples. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Mixed covolume method for parabolic problems on triangular grids

In this paper, we present a mixed covolume method for parabolic equations on triangular grids. This method use the lowest order Raviart–Thomas (R–T) mixed finite element space as the trial space. We prove the optimal order of convergence for the approximate pressure and velocity in L²-norm. Furthermore, we obtain the quasi-optimal error estimates for the approximate pressure in L^∞-norm.     

Symmetric mixed covolume methods for parabolic problems

We present a mixed covolume method for a system of first order partial differential equations resulting from the mixed formulation of the general self‐adjoint parabolic problem with a variable nondiagonal diffusion tensor. The lowest order Raviart‐Thomas mixed element space on rectangles is used. We prove the first order optimal rate of convergence for approximate pressure as well as for approximate velocity. We also prove the second order superconvergence both for approximate velocity and pressure in certain discrete norms. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 561–583, 2002     

A new mixed scheme based on variation of constants for Sobolev equation with nonlinear convection term

A new mixed scheme which combines the variation of constants and the H ¹-Galerkin mixed finite element method is constructed for nonlinear Sobolev equation with nonlinear convection term. Optimal error estimates are derived for both semidiscrete and fully discrete schemes. Finally, some numerical results are given to confirm the theoretical analysis of the proposed method.     

Error estimate in L2 of a covolume method for the generalized Stokes problem

We investigate an L²‐error estimate of a covolume scheme for the Stokes problem recently introduced by Chou (Math Comp 66 (1997), 85–104). We show the error in L² norm is of second order provided the exact velocity is in H ³ and the exact pressure is in H². © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

A posteriori error estimates for the mixed finite element method with Lagrange multipliers

We consider the mixed finite element method with Lagrange multipliers as applied to second‐order elliptic equations in divergence form with mixed boundary conditions. The corresponding Galerkin scheme is defined by using Raviart‐Thomas spaces. We develop a posteriori error analyses yielding a reliable and efficient estimate based on residuals, and a reliable and quasi‐efficient estimate based on local problems, respectively. Several numerical results illustrate the suitability of these a posteriori estimates for the adaptive computation of the discrete solutions. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

On the mixed finite element method with Lagrange multipliers

In this note we analyze a modified mixed finite element method for second‐order elliptic equations in divergence form. As a model we consider the Poisson problem with mixed boundary conditions in a polygonal domain of R ². The Neumann (essential) condition is imposed here in a weak sense, which yields the introduction of a Lagrange multiplier given by the trace of the solution on the corresponding boundary. This approach allows to handle nonhomogeneous Neumann boundary conditions, theoretically and computationally, in an alternative and usually easier way. Then we utilize the classical Babu?ka‐Brezzi theory to show that the resulting mixed variational formulation is well posed. In addition, we use Raviart‐Thomas spaces to define the associated finite element method and, applying some elliptic regularity results, we prove the stability, unique solvability, and convergence of this discrete scheme, under appropriate assumptions on the mesh sizes. Finally, we provide numerical results illustrating the performance of the algorithm for smooth and singular problems. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 192–210, 2003     

A two-grid method with expanded mixed element for nonlinear reaction-diffusion equations

Expanded mixed finite element approximation of nonlinear reaction-diffusion equations is discussed. The equations considered here are used to model the hydrologic and bio-geochemical phenomena. To linearize the mixed-method equations, we use a two-grid method involving a small nonlinear system on a coarse gird of size H and a linear system on a fine grid of size h. Error estimates are derived which demonstrate that the error is O(△t + h k+1 + H 2k+2 d/2 ) (k ≥ 1), where k is the degree of the approximating space for the primary variable and d is the spatial dimension. The above estimates are useful for determining an appropriate H for the coarse grid problems.     

On the subdomain‐Galerkin/least squares method for 2‐ and 3‐D mixed elliptic problems with reaction terms

In this article we apply the subdomain‐Galerkin/least squares method, which is first proposed by Chang and Gunzburger for first‐order elliptic systems without reaction terms in the plane, to solve second‐order non‐selfadjoint elliptic problems in two‐ and three‐dimensional bounded domains with triangular or tetrahedral regular triangulations. This method can be viewed as a combination of a direct cell vertex finite volume discretization step and an algebraic least‐squares minimization step in which the pressure is approximated by piecewise linear elements and the flux by the lowest order Raviart‐Thomas space. This combined approach has the advantages of both finite volume and least‐squares methods. Among other things, the combined method is not subject to the Ladyzhenskaya‐Babus?ka‐Brezzi condition, and the resulting linear system is symmetric and positive definite. An optimal error estimate in the H¹(Ω) × H(div; Ω) norm is derived. An equivalent residual‐type a posteriori error estimator is also given. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 738–751, 2002; Published online in Wiley InterScience (www.interscience.wiley.com); DOI 10.1002/num.10030.     

Superconvergence of mixed covolume method for elliptic problems on triangular grids

In this paper, we consider the superconvergence of a mixed covolume method on the quasi-uniform triangular grids for the variable coefficient-matrix Poisson equations. The superconvergence estimates between the solution of the mixed covolume method and that of the mixed finite element method have been obtained. With these superconvergence estimates, we establish the superconvergence estimates and the L^∞$L^{\infty }$-error estimates for the mixed covolume method for the elliptic problems. Based on the superconvergence of the mixed covolume method, under the condition that the triangulation is uniform, we construct a post-processing method for the approximate velocity which improves the order of approximation of the approximate velocity.     

线性抛物型积分微分方程的扩展混合体积元方法

1 引言 考虑线性抛物型积分微分方程初边值问题: {pt(x,t)-▽.{A(x,t)▽p(x,t) +∫t0 B(x,t,τ)▽p(x,τ)dτ}=f(x,t),(x,t)∈Ω×(0,T],(1.1) p(x,0):p0(x), x∈Ω, p(x,t)=0, (x,t)∈(a)Ω×(0,T]. 这里x=(x,y),Ω=(a,b)×(c,d),(e)Ω是区域Ω的边界,p为未知函数,A=(aij)2×2为已知的对称正定矩阵,B=(bij)2×2为已知矩阵,而且aij,bij,(aij)t(i,j=1,2)光滑有界,f∈L2(Ω).     

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.     

Optimal error analysis of Crank–Nicolson lowest-order Galerkin-mixed finite element method for incompressible miscible flow in porous media

Numerical methods for incompressible miscible flow in porous media have been studied extensively in the last several decades. In practical applications, the lowest-order Galerkin-mixed method is the most popular one, where the linear Lagrange element is used for the concentration and the lowest order Raviart–Thomas mixed element pair is used for the Darcy velocity and pressure. The existing error estimate of the method in L² -norm is in the order in spatial direction, which however is not optimal and valid only under certain extra restrictions on both time step and spatial meshes, excluding the most commonly used mesh h = h_p = h_c . This paper focuses on new and optimal error estimates of a linearized Crank–Nicolson lowest-order Galerkin-mixed finite element method (FEM), where the second-order accuracy for the concentration in both time and spatial directions is established unconditionally. The key to our optimal error analysis is an elliptic quasi-projection. Moreover, we propose a simple one-step recovery technique to obtain a new numerical Darcy velocity and pressure of second-order accuracy. Numerical results for both two and three-dimensional models are provided to confirm our theoretical analysis.     

A general mixed covolume framework for constructing conservative schemes for elliptic problems

We present a general framework for the finite volume or covolume schemes developed for second order elliptic problems in mixed form, i.e., written as first order systems. We connect these schemes to standard mixed finite element methods via a one-to-one transfer operator between trial and test spaces. In the nonsymmetric case (convection-diffusion equation) we show one-half order convergence rate for the flux variable which is approximated either by the lowest order Raviart-Thomas space or by its image in the space of discontinuous piecewise constants. In the symmetric case (diffusion equation) a first order convergence rate is obtained for both the state variable (e.g., concentration) and its flux. Numerical experiments are included.

     

A posteriori error estimator for expanded mixed hybrid methods

In this article, we construct an a posteriori error estimator for expanded mixed hybrid finite‐element methods for second‐order elliptic problems. An a posteriori error analysis yields reliable and efficient estimate based on residuals. Several numerical examples are presented to show the effectivity of our error indicators. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 330–349, 2007     

Error estimates of H^1-Galerkin mixed finite element method for Schrodinger equation

An H1-Galerkin mixed finite element method is discussed for a class of second order SchrSdinger equation. Optimal error estimates of semidiscrete schemes are derived for problems in one space dimension. At the same time, optimal error estimates are derived for fully discrete schemes. And it is showed that the H1-Galerkin mixed finite element approximations have the same rate of convergence as in the classical mixed finite element methods without requiring the LBB consistency condition.     

A two‐grid method for characteristic expanded mixed finite element solution of miscible displacement problem

A combined method consisting of mixed finite element method (MFEM) for the pressure equation and expanded mixed finite element method with characteristics(CEMFEM) for the concentration equation is presented to solve the coupled system of incompressible miscible displacement problem. To solve the resulting nonlinear system of equations efficiently, the two‐grid algorithm relegates all of the Newton‐like iterations to grids much coarser than the original one, with no loss in order of accuracy. It is shown that coarse space can be extremely coarse and our algorithm achieve asymptotically optimal approximation when the mesh sizes satisfy $H=O(h^{\frac{1}{4}})$. Numerical experiment is provided to confirm our theoretical results.     

Time discontinuous Galerkin space-time finite element method for nonlinear Sobolev equations

This article presents a complete discretization of a nonlinear Sobolev equation using space-time discontinuous Galerkin method that is discontinuous in time and continuous in space. The scheme is formulated by introducing the equivalent integral equation of the primal equation. The proposed scheme does not explicitly include the jump terms in time, which represent the discontinuity characteristics of approximate solution. And then the complexity of the theoretical analysis is reduced. The existence and uniqueness of the approximate solution and the stability of the scheme are proved. The optimalorder error estimates in L²(H¹) and L²(L²) norms are derived. These estimates are valid under weak restrictions on the space-time mesh, namely, without the condition k_n≥ch², which is necessary in traditional space-time discontinuous Galerkin methods. Numerical experiments are presented to verify the theoretical results.     

An iterative method for a Cauchy problem for the heat equation

^** Email: tomjo{at}itn.liu.se An iterative method for reconstruction of the solution to aparabolic initial boundary value problem of second order fromCauchy data is presented. The data are given on a part of theboundary. At each iteration step, a series of well-posed mixedboundary value problems are solved for the parabolic operatorand its adjoint. The convergence proof of this method in a weightedL²-space is included.     

Least‐squares method for the Oseen equation

This article studies the least‐squares finite element method for the linearized, stationary Navier–Stokes equation based on the stress‐velocity‐pressure formulation in d dimensions (d = 2 or 3). The least‐squares functional is simply defined as the sum of the squares of the L² norm of the residuals. It is shown that the homogeneous least‐squares functional is elliptic and continuous in the norm. This immediately implies that the a priori error estimate of the conforming least‐squares finite element approximation is optimal in the energy norm. The L² norm error estimate for the velocity is also established through a refined duality argument. Moreover, when the right‐hand side f belongs only to , we derive an a priori error bound in a weaker norm, that is, the norm. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1289–1303, 2016