共查询到20条相似文献,搜索用时 15 毫秒
1.
Ricardo H. Nochetto 《Numerische Mathematik》1989,54(6):601-618
Summary Consider the following quasilinear elliptic PDE, which is equivalent to a nonlinear variational inequality: –divF(u)+(u)f. Here is a singular maximal monotone graph and the nonlinear differential operator is only assumed to be monotone; surfaces of prescribed mean curvature over obstacles may thus be viewed as relevant examples. The numerical approximation proposed in this paper consists of combining continuous piecewise linear finite elements with a preliminary regularization of . The resulting scheme is shown to be quasi-optimally accurate inL
. The underlying analysis makes use of both a topological technique and a sharpL
p
-duality argument.This work was partially supported by Consiglio Nazionale delle Ricerche of Italy while the author was in residence at the Istituto di Analisi Numerica del C.N.R. di Pavia 相似文献
2.
A compact finite difference method with non-isotropic mesh is proposed for a two-dimensional fourth-order nonlinear elliptic boundary value problem. The existence and uniqueness of its solutions are investigated by the method of upper and lower solutions, without any requirement of the monotonicity of the nonlinear term. Three monotone and convergent iterations are provided for resolving the resulting discrete systems efficiently. The convergence and the fourth-order accuracy of the proposed method are proved. Numerical results demonstrate the high efficiency and advantages of this new approach. 相似文献
3.
We consider the task of resolving accurately the nth eigenpair of a generalized eigenproblem rooted in some elliptic partial differential equation (PDE), using an adaptive finite element method (FEM). Conventional adaptive FEM algorithms call a generalized eigensolver after each mesh refinement step. This is not practical in our situation since the generalized eigensolver needs to calculate n eigenpairs after each mesh refinement step, it can switch the order of eigenpairs, and for repeated eigenvalues it can return an arbitrary linear combination of eigenfunctions from the corresponding eigenspace. In order to circumvent these problems, we propose a novel adaptive algorithm that only calls a generalized eigensolver once at the beginning of the computation, and then employs an iterative method to pursue a selected eigenvalue–eigenfunction pair on a sequence of locally refined meshes. Both Picard’s and Newton’s variants of the iterative method are presented. The underlying partial differential equation (PDE) is discretized with higher-order finite elements (hp-FEM) but the algorithm also works for standard low-order FEM. The method is described and accompanied with theoretical analysis and numerical examples. Instructions on how to reproduce the results are provided. 相似文献
4.
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
5.
Luca F. Pavarino 《Numerische Mathematik》1994,69(2):185-211
Summary.
In some applications, the accuracy of the numerical solution of an
elliptic problem needs to be increased only in certain parts of the
domain. In this paper, local refinement is introduced for an overlapping
additive Schwarz algorithm for the $-version finite element method.
Both uniform and variable degree refinements are considered.
The resulting algorithm is highly parallel and scalable.
In two and three dimensions,
we prove an optimal bound for the condition number of the iteration
operator under certain hypotheses on the refinement region.
This bound is independent of the degree $, the number of
subdomains $ and the mesh size $.
In the general two dimensional case, we prove an almost optimal bound
with polylogarithmic growth in $.
Received February 20, 1993 / Revised version received January
20, 1994 相似文献
6.
Approximation theoretic results are obtained for approximation using continuous piecewise polynomials of degree p on meshes of triangular and quadrilateral elements. Estimates for the rate of convergence in Sobolev spaces , are given. The results are applied to estimate the rate of convergence when the p-version finite element method is used to approximate the -Laplacian. It is shown that the rate of convergence of the p-version is always at least that of the h-version (measured in terms of number of degrees of freedom used). If the solution is very smooth then the p-version attains an exponential rate of convergence. If the solution has certain types of singularity, the rate of convergence
of the p-version is twice that of the h-version. The analysis generalises the work of Babuska and others to the case . In addition, the approximation theoretic results find immediate application for some types of spectral and spectral element
methods.
Received August 2, 1995 / Revised version received January 26, 1998 相似文献
7.
This paper proposes a descent method to solve a class of structured monotone variational inequalities. The descent directions are constructed from the iterates generated by a prediction-correction method [B.S. He, Y. Xu, X.M. Yuan, A logarithmic-quadratic proximal prediction-correction method for structured monotone variational inequalities, Comput. Optim. Appl. 35 (2006) 19-46], which is based on the logarithmic-quadratic proximal method. In addition, the optimal step-sizes along these descent directions are identified to accelerate the convergence of the new method. Finally, some numerical results for solving traffic equilibrium problems are reported. 相似文献
8.
Ralf Kornhuber 《Numerische Mathematik》1996,72(4):481-499
Summary.
We derive globally convergent multigrid methods
for discrete elliptic
variational inequalities of the second kind
as obtained from
the approximation of related continuous
problems by piecewise linear finite elements.
The coarse grid corrections are computed
from certain obstacle problems.
The actual constraints are fixed by the
preceding nonlinear fine grid smoothing.
This new approach allows the implementation
as a classical V-cycle and preserves
the usual multigrid efficiency.
We give estimates
for the asymptotic convergence rates.
The numerical results indicate a significant improvement
as compared with previous multigrid approaches.
Received
March 26, 1994 / Revised version received September 22, 1994 相似文献
9.
The article deals with Galerkin matrices arising with finite element discretizations of the Navier–Stokes system. Usually these matrices are indefinite and nonsymmetric. They have to be preconditioned if a related linear system is to be solved efficiently by an iterative method. We consider preconditioning by a pressure mass matrix. It is shown how upper and lower bounds of the eigenvalues of a preconditioned Galerkin matrix may be found by variational arguments. 相似文献
10.
Rate of Convergence for some constraint decomposition methods for nonlinear variational inequalities 总被引:3,自引:0,他引:3
Xue-Cheng Tai 《Numerische Mathematik》2003,93(4):755-786
Summary. Some general subspace correction algorithms are proposed for a convex optimization problem over a convex constraint subset.
One of the nontrivial applications of the algorithms is the solving of some obstacle problems by multilevel domain decomposition
and multigrid methods. For domain decomposition and multigrid methods, the rate of convergence for the algorithms for obstacle
problems is of the same order as the rate of convergence for jump coefficient linear elliptic problems. In order to analyse
the convergence rate, we need to decompose a finite element function into a sum of functions from the subspaces and also satisfying
some constraints. A special nonlinear interpolation operator is introduced for decomposing the functions.
Received December 13, 2001 / Revised version received February 19, 2002 / Published online June 17, 2002
This work was partially supported by the Norwegian Research Council under projects 128224/431 and SEP-115837/431. 相似文献
11.
In this article, we present a new fully discrete finite element nonlinear Galerkin method, which are well suited to the long
time integration of the Navier-Stokes equations. Spatial discretization is based on two-grid finite element technique; time
discretization is based on Euler explicit scheme with variable time step size. Moreover, we analyse the boundedness, convergence
and stability condition of the finite element nonlinear Galerkin method. Our discussion shows that the time step constraints
of the method depend only on the coarse grid parameter and the time step constraints of the finite element Galerkin method depend on the fine grid parameter under the same convergence accuracy.
Received February 2, 1994 / Revised version received December 6, 1996 相似文献
12.
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
13.
Abdellah Bnouhachem Min Li Sheng Zhaohan 《Journal of Computational and Applied Mathematics》2010,234(12):3356-3365
In this paper, we suggest and analyze an inexact implicit method with a variable parameter for mixed variational inequalities by using a new inexactness restriction. Under certain conditions, the global convergence of the proposed method is proved. Some preliminary computational results are given to illustrate the efficiency of the new inexactness restriction. The results proved in this paper may be viewed as improvement and refinement of the previously known results. 相似文献
14.
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
15.
Summary Free boundary value problems, too complicated for formulation as a variational inequality, are broken up into two problems on overlapping regions. On one region the problem is treated as an ordinary boundary value problem; on the second region, the free boundary part of the problem is reduced to a variational inequality. By solving the two problems successively it is shown that under certain conditions the successive solutions converge to a single function that gives a solution of the original problem. Application to a filtration problem is given. 相似文献
16.
Residual-based a posteriori error estimates were derived within one unifying framework for lowest-order conforming, nonconforming,
and mixed finite element schemes in Carstensen [Numer Math 100:617–637, 2005]. Therein, the key assumption is that the conforming
first-order finite element space annulates the linear and bounded residual ℓ written . That excludes particular nonconforming finite element methods (NCFEMs) on parallelograms in that . The present paper generalises the aforementioned theory to more general situations to deduce new a posteriori error estimates,
also for mortar and discontinuous Galerkin methods. The key assumption is the existence of some bounded linear operator with some elementary properties. It is conjectured that the more general hypothesis (H1)–(H3) can be established for all
known NCFEMs. Applications on various nonstandard finite element schemes for the Laplace, Stokes, and Navier–Lamé equations
illustrate the presented unifying theory of a posteriori error control for NCFEM.
Supported by DFG Research Center MATHEON “Mathematics for key technologies” in Berlin and the German Indian Project DST-DAAD
(PPP-05). J. Hu was partially supported by National Science Foundation of China under Grant No.10601003. 相似文献
17.
18.
Alexander Ženíšek 《Numerische Mathematik》1990,58(1):51-77
Summary The study of the finite element approximation to nonlinear second order elliptic boundary value problems with discontinuous coefficients is presented in the case of mixed Dirichlet-Neumann boundary conditions. The change in domain and numerical integration are taken into account. With the assumptions which guarantee that the corresponding operator is strongly monotone and Lipschitz-continuous the following convergence results are proved: 1. the rate of convergenceO(h
) if the exact solutionuH
1 () is piecewise of classH
1+ (0<1);2. the convergence without any rate of convergence ifuH
1 () only. 相似文献
19.
In some real-world problems, the mapping of the variational inequalities does not have any explicit forms and only the function value can be evaluated or observed for given variables. In this case, if the mapping is co-coercive, the basic projection method is applicable. However, in order to determine the step size, the existing basic projection method needs to know the co-coercive modulus in advance. In practice, usually even if the mapping can be characterized co-coercive, it is difficult to evaluate the modulus, and a conservative estimation will lead an extremely slow convergence. In view of this point, this paper presents a self-adaptive projection method without knowing the co-coercive modulus. We also give a real-life example to demonstrate the practicability of the proposed method. 相似文献
20.
Wenbin Liu 《Numerische Mathematik》2000,86(3):491-506
Summary. In [13], a nonlinear elliptic equation arising from elastic-plastic mechanics is studied. A well-posed weak formulation is
established for the equation and some regularity results are further obtained for the solution of the boundary problem. In
this work, the finite element approximation of this boundary problem is examined in the framework of [13]. Some error bounds
for this approximation are initially established in an energy type quasi-norm, which naturally arises in degenerate problems
of this type and proves very useful in deriving sharper error bounds for the finite element approximation of such problems.
For sufficiently regular solutions optimal error bounds are then obtained for some fully degenerate cases in energy type norms.
Received June 12, 1998 / Revised version received June 21, 1999 / Published online June 8, 2000 相似文献