共查询到20条相似文献,搜索用时 15 毫秒
1.
Daoqi Yang 《Numerische Mathematik》2002,93(1):177-200
Summary Iterative schemes for mixed finite element methods are proposed and analyzed in two abstract formulations. The first one has
applications to elliptic equations and incompressible fluid flow problems, while the second has applications to linear elasticity
and compressible Stokes problems. These schemes are constructed through iteratively penalizing the mixed finite element scheme,
of which iterated penalty method and augmented Lagrangian method are special cases. Convergence theorems are demonstrated
in abstract formulations in Hilbert spaces, and applications to individual physical problems are considered as examples. Theoretical
analysis and computational experiments both show that the proposed schemes have very fast convergence; a few iterations are
normally enough to reduce the iterative error to a prescribed precision. Numerical examples with continuous and discontinuous
coefficients are presented. 相似文献
2.
A new technique is developed for plane or spherical moving-boundaryproblems, such as occur in freezing or melting problems. A coordinatetransformation for immobilization of the moving boundary isused. The technique involves the use of a semidiscrete PetrovGalerkinmethod with the piecewise-exponential test functions and thepiecewise-linear trial functions to approximate the timedependentgoverning equation and the use of a predictor-corrector methodto integrate the boundary motion equation. Computed resultsfor the plane or spherical moving boundary are numerically evaluatedfor comparison with results of previous authors. This methodis simple and there is no limit on the range of the parameters.We argue that the method is well-suited to a variety of movingboundaryproblems. 相似文献
3.
Yanping Chen. 《Mathematics of Computation》2008,77(263):1269-1291
In this paper, we investigate the superconvergence property of the numerical solution of a quadratic convex optimal control problem by using rectangular mixed finite element methods. The state and co-state variables are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. Some realistic regularity assumptions are presented and applied to error estimation by using an operator interpolation technique. We derive superconvergence properties for the flux functions along the Gauss lines and for the scalar functions at the Gauss points via mixed projections. Moreover, global superconvergence results are obtained by virtue of an interpolation postprocessing technique. Thus, based on these superconvergence estimates, some asymptotic exactness a posteriori error estimators are presented for the mixed finite element methods. Finally, some numerical examples are given to demonstrate the practical side of the theoretical results about superconvergence.
4.
We consider a class of mixed finite element methods for nonlinear parabolic problems over a plane domain. The finite element spaces taken are Raviart-Thomas spaces of index k, k ? 0. We obtain optimal order L2- and almost optimal order L∞-error estimates for the finite element solution and order optimal L2-error estimates for its gradient. We also derive the error estimates for the time derivatives of the solution. Our results extend those previously obtained by Johnson and Thomée for the corresponding linear problems with k ? 1. 相似文献
5.
Summary. A semidiscrete mixed finite element approximation to parabolic initial-boundary value problems is introduced and analyzed.
Superconvergence estimates for both pressure and velocity are obtained. The estimates for the errors in pressure and velocity
depend on the smoothness of the initial data including the limiting cases of data in and data in , for sufficiently large. Because of the smoothing properties of the parabolic operator, these estimates for large time levels
essentially coincide with the estimates obtained earlier for smooth solutions. However, for small time intervals we obtain
the correct convergence orders for nonsmooth data.
Received July 30, 1995 / Revised version received October 14, 1996 相似文献
6.
Summary For solving second order elliptic problems discretized on a sequence of nested mixed finite element spaces nearly optimal iterative methods are proposed. The methods are within the general framework of the product (multiplicative) scheme for operators in a Hilbert space, proposed recently by Bramble, Pasciak, Wang, and Xu [5,6,26,27] and make use of certain multilevel decomposition of the corresponding spaces for the flux variable. 相似文献
7.
In this paper, the multipoint flux mixed finite element method is used to approximate the flux of two-dimensional elliptic interface problems. Within the class of modified quasi-monotonically distributed coefficients, we derive uniformly robust residual-type a posteriori error estimators for the flux error. Based on the residual-type estimator, we further develop robust implicit and explicit recovery-type estimators through gradient recovery in H(curl) conforming finite element spaces. Numerical experiments are presented to support the theoretical results. 相似文献
8.
Mohamed Farhloul And Michel Fortin 《Numerical Methods for Partial Differential Equations》1997,13(5):445-457
In a recent work, Hiptmair [Mathematisches Institut, M9404, 1994] has constructed and analyzed a family of nonconforming mixed finite elements for second-order elliptic problems. However, his analysis does not work on the lowest order elements. In this article, we show that it is possible to construct a nonconforming mixed finite element for the lowest order case. We prove the convergence and give estimates of optimal order for this finite element. Our proof is based on the use of the properties of the so-called nonconforming bubble function to control the consistency terms introduced by the nonconforming approximation. We further establish an equivalence between this mixed finite element and the nonconforming piecewise quadratic finite element of Fortin and Soulie [J. Numer. Methods Eng., 19, 505–520, 1983]. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 445–457, 1997 相似文献
9.
Madhusmita Tripathy 《Applicable analysis》2013,92(8):1213-1231
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. 相似文献
10.
Analysis of least-squares mixed finite element methods for nonlinear nonstationary convection-diffusion problems 总被引:6,自引:0,他引:6
Dan-Ping Yang. 《Mathematics of Computation》2000,69(231):929-963
Some least-squares mixed finite element methods for convection-diffusion problems, steady or nonstationary, are formulated, and convergence of these schemes is analyzed. The main results are that a new optimal a priori error estimate of a least-squares mixed finite element method for a steady convection-diffusion problem is developed and that four fully-discrete least-squares mixed finite element schemes for an initial-boundary value problem of a nonlinear nonstationary convection-diffusion equation are formulated. Also, some systematic theories on convergence of these schemes are established.
11.
In this paper, we investigate the error estimates for the solutions of optimal control problems by mixed finite element methods. The state and costate are approximated by Raviart-Thomas mixed finite element spaces of order k and the control is approximated by piecewise polynomials of order k. Under the special constraint set, we will show that the control variable can be smooth in the whole domain. We derive error estimates of optimal order both for the state variables and the control variable. 相似文献
12.
Summary. Stabilisation methods are often used to circumvent the difficulties associated with the stability of mixed finite element methods. Stabilisation however also means an excessive amount of dissipation or the loss of nice conservation properties. It would thus be desirable to reduce these disadvantages to a minimum. We present a general framework, not restricted to mixed methods, that permits to introduce a minimal stabilising term and hence a minimal perturbation with respect to the original problem. To do so, we rely on the fact that some part of the problem is stable and should not be modified. Sections 2 and 3 present the method in an abstract framework. Section 4 and 5 present two classes of stabilisations for the inf-sup condition in mixed problems. We present many examples, most arising from the discretisation of flow problems. Section 6 presents examples in which the stabilising terms is introduced to cure coercivity problems. Received August 9, 1999 / Revised version received May 19, 2000 / Published online March 20, 2001 相似文献
13.
A mixed multiscale finite element method for elliptic problems with oscillating coefficients 总被引:5,自引:0,他引:5
The recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the large-scale structure of the solutions without resolving all the fine-scale structures. Motivated by the numerical simulation of flow transport in highly heterogeneous porous media, we propose a mixed multiscale finite element method with an over-sampling technique for solving second order elliptic equations with rapidly oscillating coefficients. The multiscale finite element bases are constructed by locally solving Neumann boundary value problems. We provide a detailed convergence analysis of the method under the assumption that the oscillating coefficients are locally periodic. While such a simplifying assumption is not required by our method, it allows us to use homogenization theory to obtain the asymptotic structure of the solutions. Numerical experiments are carried out for flow transport in a porous medium with a random log-normal relative permeability to demonstrate the efficiency and accuracy of the proposed method. 相似文献
14.
The goal of this paper is to study a mixed finite element approximation of the general convex optimal control problems governed
by quasilinear elliptic partial differential equations. The state and co-state are approximated by the lowest order Raviart-Thomas
mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a priori error estimates
both for the state variables and the control variable. Finally, some numerical examples are given to demonstrate the theoretical
results. 相似文献
15.
Neela Nataraj 《Applied mathematics and computation》2009,213(1):60-72
A mixed finite element method for approximating eigenpairs of IV order elliptic eigenvalue problems with Dirichlet boundary conditions has been given. The method can be applied to the vibration analysis of anisotropic/orthotropic/isotropic/biharmonic plates. Computer implementation procedures for this mixed method are given along with the results of numerical experiments. 相似文献
16.
Mixed control-state constraints are used as a relaxation of originally state constrained optimal control problems for partial
differential equations to avoid the intrinsic difficulties arising from measure-valued multipliers in the case of pure state
constraints. In particular, numerical solution techniques known from the pure control constrained case such as active set
strategies and interior-point methods can be used in an appropriately modified way. However, the residual-type a posteriori
error estimators developed for the pure control constrained case can not be applied directly. It is the essence of this paper
to show that instead one has to resort to that type of estimators known from the pure state constrained case. Up to data oscillations
and consistency error terms, they provide efficient and reliable estimates for the discretization errors in the state, a regularized
adjoint state, and the control. A documentation of numerical results is given to illustrate the performance of the estimators. 相似文献
17.
18.
A semidiscrete finite element scheme for the approximation of the spatial temperature change field is presented. The method yields a better order of convergence than the conventional use of linear elements. 相似文献
19.
Summary.
We consider the mixed formulation for the
elasticity problem and the limiting
Stokes problem in ,
.
We derive a set of sufficient conditions under which families of
mixed finite element spaces
are simultaneously stable with respect to the mesh size
and, subject to a
maximum loss of
,
with respect to the polynomial
degree .
We obtain asymptotic
rates of convergence that are optimal up to
in the
displacement/velocity and up to
in the
"pressure", with
arbitrary
(both rates being
optimal with respect to
). Several choices of
elements are discussed with reference to
properties desirable in the
context of the -version.
Received
March 4, 1994 / Revised version received February 12, 1995 相似文献
20.
On finite element methods for plasticity problems 总被引:3,自引:0,他引:3
Claes Johnson 《Numerische Mathematik》1976,26(1):79-84
Summary We prove an error estimate for an incremental finite element method for obtaining approximations to the stresses in an elastic-perfectly plastic body. We also comment on the limit load problem. 相似文献