基于Riesz-表示算子非协调稳定化有限元法

Following the framework of the finite element methods based on Riesz-representing operators developed by Duan Huoyuan in 1997,through discrete Riesz representing-operators on some virtual(non-) conforming finite-dimensional subspaces,a stabilization formulation is presented for the Stokes problem by employing nonconforming elements.This formulation is uniformly coercive and not subject to the Babu ka-Brezzi condition,and the resulted linear algebraic system is positive definite with the spectral condition number O(h-2). Quasi-optimal error bounds are obtained,which is consistent with the interpolation properties of the finite elements used.     

Mixed finite element methods based on Riesz-representing operators for the shell problem

1. IntroductionAs far as the shell problem is concerned, [l] established a mixed formulation in the clas-sical W that the K-ellipticity and the lnfSup condition are introduced. Unfortunately it isvery dndcult to construct ndxed elemellts simultaneously satisfying the K-ellipticity and theInfSup conditionI2], which are indeed prerequisites Of the stability and the convergence. Inaddition, the indefiniteness of the resulting linear algebraic system complicates the solutionalgorithIn.ffecentl…     

Review and complements on mixed-hybrid finite element methods for fluid flows

Mixed and hybrid finite element methods for the resolution of a wide range of linear and nonlinear boundary value problems (linear elasticity, Stokes problem, Navier–Stokes equations, Boussinesq equations, etc.) have known a great development in the last few years. These methods allow simultaneous computation of the original variable and its gradient, both of them being equally accurate. Moreover, they have local conservation properties (conservation of the mass and the momentum) as in the finite volume methods.The purpose of this paper is to give a review on some mixed finite elements developed recently for the resolution of Stokes and Navier–Stokes equations, and the linear elasticity problem. Further developments for a quasi-Newtonian flow obeying the power law are presented.     

Some remarks on the flux-free finite element method for immiscible two-fluid flows

We give some theoretical considerations on the the flux-free finite element method for the generalized Stokes interface problem arising from the immiscible two-fluid flow problems. In the flux-free finite element method, the flux constraint is posed as another Lagrange multiplier to keep the zero-flux on the interface. As a result, the mass of each fluid is expected to be preserved at every time step. We first study the effect of discontinuous coefficients (viscosity and density) on the error of the standard finite element approximations very carefully. Then, the analysis is extended to the flux-free finite element method.     

Pressure projection stabilized finite element method for Stokes problem with nonlinear slip boundary conditions

In this paper, we consider the pressure projection stabilized finite element method for the Stokes problem with nonlinear slip boundary conditions whose variational formulation is the variational inequality problem of the second kind with the Stokes operator. The H¹ and L² error estimates for the velocity and the L² error estimate for the pressure are obtained. Finally, the numerical results are displayed to verify the theoretical analysis.     

Optimal control of the convection-diffusion equation using stabilized finite element methods

In this paper we analyze the discretization of optimal control problems governed by convection-diffusion equations which are subject to pointwise control constraints. We present a stabilization scheme which leads to improved approximate solutions even on corse meshes in the convection dominated case. Moreover, the in general different approaches “optimize-then- discretize” and “discretize-then-optimize” coincide for the proposed discretization scheme. This allows for a symmetric optimality system at the discrete level and optimal order of convergence.     

Approximation and eigenvalue extrapolation of Stokes eigenvalue problem by nonconforming finite element methods

In this paper we analyze the stream function-vorticity-pressure method for the Stokes eigenvalue problem. Further, we obtain full order convergence rate of the eigenvalue approximations for the Stokes eigenvalue problem based on asymptotic error expansions for two nonconforming finite elements, Q ₁^rot and EQ ₁^rot. Using the technique of eigenvalue error expansion, the technique of integral identities and the extrapolation method, we can improve the accuracy of the eigenvalue approximations. This project is supported in part by the National Natural Science Foundation of China (10471103) and is subsidized by the National Basic Research Program of China under the grant 2005CB321701.     

Half-inverse problem for diffusion operators on the finite interval

The potential function q(x) in the regular and singular Sturm-Liouville problem can be uniquely determined from two spectra. Inverse problem for diffusion operator given at the finite interval eigenvalues, normal numbers also on two spectra are solved. Half-inverse spectral problem for a Sturm-Liouville operator consists in reconstruction of this operator by its spectrum and half of the potential. In this study, by using the Hochstadt and Lieberman's method we show that if q(x) is prescribed on , then only one spectrum is sufficient to determine q(x) on the interval for diffusion operator.     

A note on the singular linear system of the generalized finite element methods

We characterize the kernel of the global stiffness matrix in the singular linear system of the generalized finite element methods (GFEM) which uses the classical finite element (FE) shape functions and local approximation space of harmonic polynomials.     

Global superconvergence of finite element methods for parabolic inverse problems

In this article we transform a large class of parabolic inverse problems into a nonclassical parabolic equation whose coefficients consist of trace type functionals of the solution and its derivatives subject to some initial and boundary conditions. For this nonclassical problem, we study finite element methods and present an immediate analysis for global superconvergence for these problems, on basis of which we obtain a posteriori error estimators. This research was supported in part by the Shahid Beheshti University, the National Basic Research Program of China (2007CB814906), the National Natural Science Foundation of China (10471103 and 10771158), Social Science Foundation of the Ministry of Education of China (Numerical methods for convertible bonds, 06JA630047), Tianjin Natural Science Foundation (07JCYBJC14300).     

Nonconforming cell boundary element methods for elliptic problems on triangular mesh

The nonconforming cell boundary element (CBE) methods are proposed. The methods are designed in such a way that they enjoy the mass conservation at the element level and the normal component of fluxes at inter-element boundaries are continuous for unstructured triangular meshes. Normal flux continuity and the optimal order error estimates in a broken H¹ norm for the P₁ method are established, which are completion of authors' earlier works [Y. Jeon, D. Sheen, Analysis of a cell boundary element method, Adv. Comput. Math. 22 (3) (2005) 201–222; Y. Jeon, E.-J. Park, D. Sheen, A cell boundary element method for elliptic problems, Numer. Methods Partial Differential Equations 21 (3) (2005) 496–511]. Moreover, two second order methods (the and modified methods) and a multiscale CBE method are constructed and numerical experiments are performed. Numerical results show feasibility and effectiveness of the CBE methods.     

A stabilized finite element method based on local polynomial pressure projection for the stationary Navier–Stokes equations

This article considers a stabilized finite element approximation for the branch of nonsingular solutions of the stationary Navier–Stokes equations based on local polynomial pressure projection by using the lowest equal-order elements. The proposed stabilized method has a number of attractive computational properties. Firstly, it is free from stabilization parameters. Secondly, it only requires the simple and efficient calculation of Gauss integral residual terms. Thirdly, it can be implemented at the element level. The optimal error estimate is obtained by the standard finite element technique. Finally, comparison with other methods, through a series of numerical experiments, shows that this method has better stability and accuracy.     

The coupling of boundary integral and finite element methods for the nonstationary exterior flow

In this article, we represent a new numerical method for solving the nonstationary Navier–Stokes equations in an unbounded domain. The technique consists of coupling the boundary integral and the finite element method. The variational formulation and the well-posedness of the coupling method are obtained. The convergence and optimal error estimates for the approximate solution are provided. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 549–565, 1998     

Two-level multiscale finite element methods for the steady navier-stokes problem

In this article, on the basis of two-level discretizations and multiscale finite element method, two kinds of finite element algorithms for steady Navier-Stokes problem are presented and discussed. The main technique is first to use a standard finite element discretization on a coarse mesh to approximate low frequencies, then to apply the simple and Newton scheme to linearize discretizations on a fine grid. At this process, multiscale finite element method as a stabilized method deals with the lowest equal-order finite element pairs not satisfying the inf-sup condition. Under the uniqueness condition, error analyses for both algorithms are given. Numerical results are reported to demonstrate the effectiveness of the simple and Newton scheme.     

Postprocessing and higher order convergence for the mixed finite element approximations of the Stokes eigenvalue problems

In this paper we propose a method for improving the convergence rate of the mixed finite element approximations for the Stokes eigenvalue problem. It is based on a postprocessing strategy that consists of solving an additional Stokes source problem on an augmented mixed finite element space which can be constructed either by refining the mesh or by using the same mesh but increasing the order of the mixed finite element space. Dedicated to Ivan Hlaváček on the occasion of his 75th birthday     

Least‐squares mixed finite element methods for the incompressible Navier‐Stokes equations

Least‐squares mixed finite element schemes are formulated to solve the evolutionary Navier‐Stokes equations and the convergence is analyzed. We recast the Navier‐Stokes equations as a first‐order system by introducing a vorticity flux variable, and show that a least‐squares principle based on L² norms applied to this system yields optimal discretization error estimates. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 441–453, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10015     

Connections between discontinuous Galerkin and nonconforming finite element methods for the Stokes equations

We study a discontinuous Galerkin finite element method (DGFEM) for the Stokes equations with a weak stabilization of the viscous term. We prove that, as the stabilization parameter γ tends to infinity, the solution converges at speed γ^?1 to the solution of some stable and well‐known nonconforming finite element methods (NCFEM) for the Stokes equations. In addition, we show that an a posteriori error estimator for the DGFEM‐solution based on the reconstruction of a locally conservative H(div, Ω)‐tensor tends at the same speed to a classical a posteriori error estimator for the NCFEM‐solution. These results can be used to affirm the robustness of the DGFEM‐method and also underline the close relationship between the two approaches. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Optimal error estimates of mixed finite element methods for a fourth-order nonlinear elliptic problem

A mixed finite element method is developed for a nonlinear fourth-order elliptic problem. Optimal L² error estimates are proved by using a special interpolation operator on the standard tensor-product finite elements of order k?1. Then two iterative schemes are presented and proved to keep the same optimal error estimates. Three numerical examples are provided to support the theoretical analysis.     

A posteriori error analysis of the discontinuous finite element methods for first order hyperbolic problems

In this paper, we consider the a posteriori error analysis of discontinuous Galerkin finite element methods for the steady and nonsteady first order hyperbolic problems with inflow boundary conditions. We establish several residual-based a posteriori error estimators which provide global upper bounds and a local lower bound on the error. Further, for nonsteady problem, we construct a fully discrete discontinuous finite element scheme and derive the a posteriori error estimators which yield global upper bound on the error in time and space. Our a posteriori error analysis is based on the mesh-dependent a priori estimates for the first order hyperbolic problems. These a posteriori error analysis results can be applied to develop the adaptive discontinuous finite element methods.     

Superconvergence of H 1-Galerkin mixed finite element methods for parabolic problems

In this article, we study the semidiscrete H ¹-Galerkin mixed finite element method for parabolic problems over rectangular partitions. The well-known optimal order error estimate in the L ²-norm for the flux is of order 𝒪(h ^k+1) (SIAM J. Numer. Anal. 35 (2), (1998), pp. 712–727), where k ≥ 1 is the order of the approximating polynomials employed in the Raviart–Thomas element. We derive a superconvergence estimate of order 𝒪(h ^k+3) between the H ¹-Galerkin mixed finite element approximation and an appropriately defined local projection of the flux variable when k ≥ 1. A the new approximate solution for the flux with superconvergence of order 𝒪(h ^k+3) is realized via a postprocessing technique using local projection methods.