共查询到20条相似文献,搜索用时 10 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.
Gabriel N. Gatica Antonio Márquez 《Journal of Computational and Applied Mathematics》2009,231(2):526-540
In this paper we introduce and analyze a new augmented mixed finite element method for linear elasticity problems in 3D. Our approach is an extension of a technique developed recently for plane elasticity, which is based on the introduction of consistent terms of Galerkin least-squares type. We consider non-homogeneous and homogeneous Dirichlet boundary conditions and prove that the resulting augmented variational formulations lead to strongly coercive bilinear forms. In this way, the associated Galerkin schemes become well posed for arbitrary choices of the corresponding finite element subspaces. In particular, Raviart-Thomas spaces of order 0 for the stress tensor, continuous piecewise linear elements for the displacement, and piecewise constants for the rotation can be utilized. Moreover, we show that in this case the number of unknowns behaves approximately as 9.5 times the number of elements (tetrahedrons) of the triangulation, which is cheaper, by a factor of 3, than the classical PEERS in 3D. Several numerical results illustrating the good performance of the augmented schemes are provided. 相似文献
3.
Serge Nicaise Juliette Venel 《Journal of Computational and Applied Mathematics》2011,235(14):4272-4282
We perform the a posteriori error analysis of residual type of transmission problem with sign changing coefficients. According to Bonnet-BenDhia et al. (2010) [9], if the contrast is large enough, the continuous problem can be transformed into a coercive one. We further show that a similar property holds for the discrete problem for any regular meshes, extending the framework from Bonnet-BenDhia et al. [9]. The reliability and efficiency of the proposed estimator are confirmed by some numerical tests. 相似文献
4.
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-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. 相似文献
5.
Summary We consider some equilibrium finite element methods for two-dimensional elasticity problems. The stresses and the displacements are approximated by using piecewise linear functions. We establishL
2-estimates of orderO(h
2) for both stresses and displacements. 相似文献
6.
Kwang-Yeon Kim Hyung-Chun Lee 《Journal of Computational and Applied Mathematics》2010,235(1):186-2912
In this work we derive and analyze a posteriori error estimators for low-order nonconforming finite element methods of the linear elasticity problem on both triangular and quadrilateral meshes, with hanging nodes allowed for local mesh refinement. First, it is shown that equilibrated Neumann data on interelement boundaries are simply given by the local weak residuals of the numerical solution. The first error estimator is then obtained by applying the equilibrated residual method with this set of Neumann data. From this implicit estimator we also derive two explicit error estimators, one of which is similar to the one proposed by Dörfler and Ainsworth (2005) [24] for the Stokes problem. It is established that all these error estimators are reliable and efficient in a robust way with respect to the Lamé constants. The main advantage of our error estimators is that they yield guaranteed, i.e., constant-free upper bounds for the energy-like error (up to higher order terms due to data oscillation) when a good estimate for the inf-sup constant is available, which is confirmed by some numerical results. 相似文献
7.
This paper deals with convergence analysis and applications of a Zienkiewicz-type (Z-type) triangular element, applied to fourth-order partial differential equations. For the biharmonic problem we prove the order of convergence by comparison to a suitable modified Hermite triangular finite element. This method is more natural and it could be applied to the corresponding fourth-order eigenvalue problem. We also propose a simple postprocessing method which improves the order of convergence of finite element eigenpairs. Thus, an a posteriori analysis is presented by means of different triangular elements. Some computational aspects are discussed and numerical examples are given. 相似文献
8.
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
9.
G. Lube 《Numerische Mathematik》1992,61(1):335-357
Summary We extend the analysis of the streamline diffusion finite element method to quasilinear elliptic problems of second order. An existence theorem and error estimates are given in the case of branches of nonsingular solutions following a recent abstract approach in [12, 13, 26]. 相似文献
10.
Finite element approximation of multi-scale elliptic problems using patches of elements 总被引:1,自引:0,他引:1
Roland Glowinski Jiwen He Alexei Lozinski Jacques Rappaz Joël Wagner 《Numerische Mathematik》2005,101(4):663-687
In this paper we present a method for the numerical solution of elliptic problems with multi-scale data using multiple levels
of not necessarily nested grids. The method consists in calculating successive corrections to the solution in patches whose
discretizations are not necessarily conforming. This paper provides proofs of the results published earlier (see C. R. Acad.
Sci. Paris, Ser. I 337 (2003) 679–684), gives a generalization of the latter to more than two domains and contains extensive
numerical illustrations. New results including the spectral analysis of the iteration operator and a numerical method to evaluate
the constant of the strengthened Cauchy-Buniakowski-Schwarz inequality are presented.
Supported by CTI Project 6437.1 IWS-IW. 相似文献
11.
Summary Finite element approximations of the eigenpairs of differential operators are computed as eigenpairs of matrices whose elements involve integrals which must be evaluated by numerical integration. The effect of this numerical integration on the eigenvalue and eigenfunction error is estimated. Specifically, for 2nd order selfadjoint eigenvalue problems we show that finite element approximations with quadrature satisfy the well-known estimates for approximations without quadrature, provided the quadrature rules have appropriate degrees of precision.The work of this author was partially supported by the National Science Foundation under Grant DMS-84-10324 相似文献
12.
Summary.
The paper deals with the finite element analysis of second
order elliptic eigenvalue problems when the approximate domains
are not subdomains of the original domain
and when at the same time numerical integration is used for computing the
involved bilinear forms. The considerations are restricted to piecewise
linear approximations. The optimum rate of convergence
for approximate
eigenvalues is obtained provided that a quadrature formula of first
degree of precision is used. In the case of a simple exact eigenvalue
the optimum rate of convergence
for approximate eigenfunctions in the
-norm is proved while in the
-norm an
almost optimum rate of convergence (i.e. near to
is achieved. In both
cases a quadrature formula of first degree of precision is used.
Quadrature formulas with degree of precision equal to zero are also
analyzed and in the case when the exact eigenfunctions belong only to
the convergence
without the rate of convergence is proved. In the case of
a multiple exact eigenvalue the approximate eigenfunctions are compard
(in contrast to standard considerations) with linear combinations of
exact eigenfunctions with coefficients not depending on the mesh
parameter .
Received September 18, 1993 / Revised
version received September 26, 1994 相似文献
13.
Helena Zarin 《Journal of Computational and Applied Mathematics》2009,231(2):626-636
We study convergence properties of a numerical method for convection-diffusion problems with characteristic layers on a layer-adapted mesh. The method couples standard Galerkin with an h-version of the nonsymmetric discontinuous Galerkin finite element method with bilinear elements. In an associated norm, we derive the error estimate as well as the supercloseness result that are uniform in the perturbation parameter. Applying a post-processing operator for the discontinuous Galerkin method, we construct a new numerical solution with enhanced convergence properties. 相似文献
14.
Two-grid finite volume element discretization techniques, based on two linear conforming finite element spaces on one coarse
and one fine grid, are presented for the two-dimensional second-order non-selfadjoint and indefinite linear elliptic problems
and the two-dimensional second-order nonlinear elliptic problems. With the proposed techniques, solving the non-selfadjoint
and indefinite elliptic problem on the fine space is reduced into solving a symmetric and positive definite elliptic problem
on the fine space and solving the non-selfadjoint and indefinite elliptic problem on a much smaller space; solving a nonlinear
elliptic problem on the fine space is reduced into solving a linear problem on the fine space and solving the nonlinear elliptic
problem on a much smaller space. Convergence estimates are derived to justify the efficiency of the proposed two-grid algorithms.
A set of numerical examples are presented to confirm the estimates.
The work is supported by the National Natural Science Foundation of China (Grant No: 10601045). 相似文献
15.
Jin-ping ZengHai-xiong Yu 《Journal of Computational and Applied Mathematics》2012,236(7):1993-2004
We are concerned with the semilinear elliptic problems. We first investigate the L2-error estimate for the lumped mass finite element method. We then use the cascadic multigrid method to solve the corresponding discrete problem. On the basis of the finite element error estimates, we prove the optimality of the proposed multigrid method. We also report some numerical results to support the theory. 相似文献
16.
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. 相似文献
17.
P. Le Tallec 《Numerische Mathematik》1982,38(3):365-382
Summary This paper considers the problems of minimizing Gateaux-differentiable functionals over subsets of real Banach spaces defined by a non-linear equality constraint. The existence of a Lagrange multiplier is proved, together with approximation results on the constrained subset, provided a nonlinear compatibility condition, generalizing the classical inf-sup condition, is satisfied. These ideas are applied to equilibrium problems in incompressible finite elasticity and lead to convergence results for these problems. 相似文献
18.
Nonconforming mixed finite element approximation to the stationary Navier-Stokes equations on anisotropic meshes 总被引:1,自引:0,他引:1
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. 相似文献
19.
Jason S. Howell 《Journal of Computational and Applied Mathematics》2009,231(2):780-792
In this work a finite element method for a dual-mixed approximation of Stokes and nonlinear Stokes problems is studied. The dual-mixed structure, which yields a twofold saddle point problem, arises in a formulation of this problem through the introduction of unknown variables with relevant physical meaning. The method approximates the velocity, its gradient, and the total stress tensor, but avoids the explicit computation of the pressure, which can be recovered through a simple postprocessing technique. This method improves an existing approach for these problems and uses Raviart-Thomas elements and discontinuous piecewise polynomials for approximating the unknowns. Existence, uniqueness, and error results for the method are given, and numerical experiments that exhibit the reduced computational cost of this approach are presented. 相似文献
20.
Changhua Yu 《Journal of Computational and Applied Mathematics》2011,236(6):1055-1068
In this paper, the semi-discrete and full discrete biquadratic finite volume element schemes based on optimal stress points for a class of parabolic problems are presented. Optimal order error estimates in H1 and L2 norms are derived. In addition, the superconvergences of numerical gradients at optimal stress points are also discussed. A numerical experiment confirms some results of theoretical analysis. 相似文献