非定常Navier-Stokes方程的质量集中各向异性非协调有限元逼近

该文将一个低阶Crouzeix-Raviart型非协调三角形元应用到非定常Navier-Stokes方程,给出了其质量集中有限元逼近格式.在不需要传统Ritz-Volterra投影下,通过引入两个辅助有限元空间对边界进行估计的技巧,在各向异性网格下导出了速度的L~2模和能量模及压力的L~2模的误差估计.     

Nonconforming mixed finite element approximation to the stationary Navier-Stokes equations on anisotropic meshes

The main aim of this paper is to study the error estimates of a rectangular nonconforming finite element for the stationary Navier-Stokes equations under anisotropic meshes. That is, the nonconforming rectangular element is taken as approximation space for the velocity and the piecewise constant element for the pressure. The convergence analysis is presented and the optimal error estimates both in a broken H¹-norm for the velocity and in an L²-norm for the pressure are derived on anisotropic meshes.     

An approximation of incompressible miscible displacement in porous media by mixed finite element method and characteristics-mixed finite element method

An approximation scheme is defined for incompressible miscible displacement in porous media. This scheme is constructed by using two methods. Standard mixed finite element is used for the Darcy velocity equation. A characteristics-mixed finite element method is presented for the concentration equation. Characteristic approximation is applied to handle the convection part of the concentration equation, and a lowest-order mixed finite element spatial approximation is adopted to deal with the diffusion part. Thus, the scalar unknown concentration and the diffusive flux can be approximated simultaneously. In order to derive the optimal L²$L^2$-norm error estimates, a post-processing step is included in the approximation to the scalar unknown concentration. This scheme conserves mass globally; in fact, on the discrete level, fluid is transported along the approximate characteristics. Numerical experiments are presented finally to validate the theoretical analysis.     

A class of nonconforming quadrilateral finite elements for incompressible flow

This paper focuses on the low-order nonconforming rectangular and quadrilateral finite elements approximation of incompressible flow.Beyond the previous research works,we propose a general strategy to construct the basis functions.Under several specific constraints,the optimal error estimates are obtained,i.e.,the first order accuracy of the velocities in H1-norm and the pressure in L2-norm,as well as the second order accuracy of the velocities in L2-norm.Besides,we clarify the differences between rectangular and quadrilateral finite element approximation.In addition,we give several examples to verify the validity of our error estimates.     

Finite element method for some nonlinear integro-differential equations

Abstract. This paper studies the finite element method for some nonlinear hyperbolic partial differential equations with memory and dampling terms. A Crank-Nicolson approximation for this kind of equations is presented. By using the elliptic Ritz Volterra projection,the analysis of the error estimates for the finite element numerical solutions and the optimal H1-norm error estimate are demonstrated.     

A posteriori error estimates for mixed finite element approximations of elliptic problems

We derive residual based a posteriori error estimates of the flux in L ²-norm for a general class of mixed methods for elliptic problems. The estimate is applicable to standard mixed methods such as the Raviart–Thomas–Nedelec and Brezzi–Douglas–Marini elements, as well as stabilized methods such as the Galerkin-Least squares method. The element residual in the estimate employs an elementwise computable postprocessed approximation of the displacement which gives optimal order.     

ERROR ESTIMATES FOR SPARSE OPTIMAL CONTROL PROBLEMS BY PIECEWISE LINEAR FINITE ELEMENT APPROXIMATION

Optimization problems with L¹-control cost functional subject to an elliptic partial differential equation(PDE)are considered.However,different from the finite dimensiona l¹-regularization optimization,the resulting discretized L¹norm does not have a decoupled form when the standard piecewise linear finite element is employed to discretize the continuous problem.A common approach to overcome this difficulty is employing a nodal quadrature formula to approximately discretize the L¹-norm.In this paper,a new discretized scheme for the L¹-norm is presented.Compared to the new discretized scheme for L¹-norm with the nodal quadrature formula,the advantages of our new discretized scheme can be demonstrated in terms of the order of approximation.Moreover,finite element error estimates results for the primal problem with the new discretized scheme for the L¹-norm are provided,which confirms that this approximation scheme will not change the order of error estimates.To solve the new discretized problem,a symmetric Gauss-Seidel based majorized accelerated block coordinate descent(sGS-mABCD)method is introduced to solve it via its dual.The proposed sGS-mABCD algorithm is illustrated at two numerical examples.Numerical results not only confirm the finite element error estimates,but also show that our proposed algorithm is efficient.     

New a posteriori L∞(L2) and L2(L2)-error estimates of mixed finite element methods for general nonlinear parabolic optimal control problems

We study new a posteriori error estimates of the mixed finite element methods for general optimal control problems governed by nonlinear parabolic equations. The state and the co-state are discretized by the high order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a posteriori error estimates in L^∞(J; L²Ω)-norm and L²(J; L²Ω)-norm for both the state, the co-state and the control approximation. Such estimates, which seem to be new, are an important step towards developing a reliable adaptive mixed finite element approximation for optimal control problems. Finally, the performance of the posteriori error estimators is assessed by two numerical examples.     

Optimal convergence analysis of an immersed interface finite element method

We analyze an immersed interface finite element method based on linear polynomials on noninterface triangular elements and piecewise linear polynomials on interface triangular elements. The flux jump condition is weakly enforced on the smooth interface. Optimal error estimates are derived in the broken H ¹-norm and L ²-norm.     

Optimal L ∞(L 2)-Error Estimates for the DG Method Applied to Nonlinear Convection–Diffusion Problems with Nonlinear Diffusion

This article is concerned with the analysis of the discontinuous Galerkin finite element method (DGFEM) applied to the space semidiscretization of a nonstationary convection–diffusion problem with nonlinear convection and nonlinear diffusion. Optimal estimates in the L ^∞(L ²)-norm are derived for the symmetric interior penalty (SIPG) scheme in two dimensions. The error analysis is carried out for nonconforming triangular meshes under the assumption that the exact solution of the problem and the solution of a linearized elliptic dual problem are sufficiently regular.     

Convergence and superconvergence analysis of an anisotropic nonconforming finite element methods for singularly perturbed reaction-diffusion problems

The numerical approximation by a lower order anisotropic nonconforming finite element on appropriately graded meshes are considered for solving singular perturbation problems. The quasi-optimal order error estimates are proved in the ε-weighted H¹-norm valid uniformly, up to a logarithmic factor, in the singular perturbation parameter. By using the interpolation postprocessing technique, the global superconvergent error estimates in ε-weighted H¹-norm are obtained. Numerical experiments are given to demonstrate validity of our theoretical analysis.     

Superconvergence in the projected-shear plate-bending finite element method

A projected-shear finite element method for periodic Reissner–Mindlin plate model are analyzed for rectangular meshes. A projection operator is applied to the shear stress term in the bilinear form. Optimal error estimates in the L²-norm, the H¹-norm, and the energy norm for both displacement and rotations are established and gradient superconvergence along the Gauss lines is justified in some weak senses. All the convergence and superconvergence results are uniform with respect to the thickness parameter t. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 367–386, 1998     

A coupling method based on new MFE and FE for fourth-order parabolic equation

In this article, a coupling method of new mixed finite element (MFE) and finite element (FE) is proposed and analyzed for fourth-order parabolic partial differential equation. First, the fourth-order parabolic equation is split into the coupled system of second-order equations. Then, an equation is solved by finite element method, the other equation is approximated by the new mixed finite element method, whose flux belongs to the square integrable space replacing the classical H(div;Ω) space. The stability for fully discrete scheme is derived, and both semi-discrete and fully discrete error estimates are obtained. Moreover, the optimal a priori error estimates in L ² and H ¹-norm for both the scalar unknown u and the diffusion term γ and a priori error estimate in (L ²)²-norm for its flux σ are derived. Finally, some numerical results are provided to validate our theoretical analysis.     

A second order nonconforming rectangular finite element method for approximating Maxwell’s equations

The main objective of this paper is to present a new rectangular nonconforming finite element scheme with the second order convergence behavior for approximation of Maxwell’s equations.Then the corresponding optimal error estimates are derived.The difficulty in construction of this finite element scheme is how to choose a compatible pair of degrees of freedom and shape function space so as to make the consistency error due to the nonconformity of the element being of order O(h 3 ) ,properly one order higher than that of its interpolation error O(h 2 ) in the broken energy norm,where h is the subdivision parameter tending to zero.     

The modified method of characteristics with mixed finite element domain decomposition procedures for the transient behavior of a semiconductor device

For the transient behavior of a semiconductor device, the modified method of characteristics with mixed finite element domain decomposition procedures applicable to parallel arithmetic is put forward. The electric potential equation is described by the mixed finite element method, and the electric, hole concentration and heat conduction equations are treated by the modified method of characteristics finite element domain decomposition methods. Some techniques, such as calculus of variations, domain decomposition, characteristic method, energy method, negative norm estimate and prior estimates and techniques are employed. Optimal order estimates in L² norm are derived for the error in the approximation solution. Thus the well‐known theoretical problem has been thoroughly and completely solved.© 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 353–368 2012     

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.     

Optimal error estimates and modified energy conservation identities of the ADI-FDTD scheme on staggered grids for 3D Maxwell’s equations

This paper is concerned with the optimal error estimates and energy conservation properties of the alternating direction implicit finite-difference time-domain(ADI-FDTD) method which is a popular scheme for solving the 3D Maxwell’s equations.Precisely,for the case with a perfectly electric conducting(PEC) boundary condition we establish the optimal second-order error estimates in both space and time in the discrete H 1-norm for the ADI-FDTD scheme,and prove the approximate divergence preserving property that if the divergence of the initial electric and magnetic fields are zero,then the discrete L 2-norm of the discrete divergence of the ADI-FDTD solution is approximately zero with the second-order accuracy in both space and time.The key ingredient is two new discrete modified energy norms which are second-order in time perturbations of two new energy conservation laws for the Maxwell’s equations introduced in this paper.Furthermore,we prove that,in addition to two known discrete modified energy identities which are second-order in time perturbations of two known energy conservation laws,the ADI-FDTD scheme also satisfies two new discrete modified energy identities which are second-order in time perturbations of the two new energy conservation laws.This means that the ADI-FDTD scheme is unconditionally stable under the four discrete modified energy norms.Experimental results which confirm the theoretical results are presented.     

An optimal order error estimate for an upwind discretization of the Navier—Stokes equations

We analyze a finite-element approximation of the stationary incompressible Navier–Stokes equations in primitive variables. This approximation is based on the nonconforming P₁/P₀ element pair of Crouzeix/Raviart and a special upwind discretization of the convective term. An optimal error estimate in a discrete H¹-norm for the velocity and in the L²-norm for the pressure is proved. Some numerical results are presented. © 1996 John Wiley & Sons, Inc.     

Anisotropic Wilson element with conforming finite element approximation for a coupled continuum pipe‐flow/Darcy model in Karst aquifers

Numerical method for a coupled continuum pipe‐flow/Darcy model describing flow in porous media with an embedded conduit pipe is considered. Wilson element on anisotropic mesh is used to solve the Darcy equation on porous matrix. The existence and uniqueness of the approximation solution are obtained. Optimal error estimates in L² and H¹ norms are established independent of the regularity condition on the mesh. Numerical examples show the efficiency of the presented scheme. With the same number of nodal points, the results using Wilson element on anisotropic mesh are much better than those of the same element and Q₁ element on regular mesh. Copyright © 2014 John Wiley & Sons, Ltd.     

A priori error estimates of finite volume method for nonlinear optimal control problem

In this paper, we study a priori error estimates for the finite volume element approximation of nonlinear optimal control problem. The schemes use discretizations based on a finite volume method. For the variational inequality, we use the method of the variational discretization concept to obtain the control. Under some reasonable assumptions, we obtain some optimal order error estimates. The approximate order for the state, costate and control variables is O(h ²) or \(O\left( {{h^2}\sqrt {\left| {\ln h} \right|} } \right)\) in the sense of L ²-norm or L ^∞-norm. A numerical experiment is presented to test the theoretical results. Finally, we give some conclusions and future works.