共查询到20条相似文献,搜索用时 0 毫秒
1.
Daisuke Furihata 《Numerische Mathematik》2001,87(4):675-699
Summary. We propose a stable and conservative finite difference scheme to solve numerically the Cahn-Hilliard equation which describes
a phase separation phenomenon. Numerical solutions to the equation is hard to obtain because it is a nonlinear and nearly
ill-posed problem. We design a new difference scheme based on a general strategy proposed recently by Furihata and Mori. The
new scheme inherits characteristic properties, the conservation of mass and the decrease of the total energy, from the equation.
The decrease of the total energy implies boundedness of discretized Sobolev norm of the solution. This in turn implies, by
discretized Sobolev's lemma, boundedness of max norm of the solution, and hence the stability of the solution. An error estimate
for the solution is obtained and the order is . Numerical examples demonstrate the effectiveness of the proposed scheme.
Received July 22, 1997 / Revised version received October 19, 1999 / Published online August 2, 2000 相似文献
2.
3.
C.V. Pao 《Numerische Mathematik》1998,79(2):261-281
This paper is concerned with numerical methods for a finite difference system of reaction-diffusion-convection equation under
nonlinear boundary condition. Various monotone iterative methods are presented, and each of these methods leads to an existence-comparison
theorem as well as a computational algorithm for numerical solutions. The monotone property of the iterations gives improved
upper and lower bounds of the solution in each iteration, and the rate of convergence of the iterations is either quadratic
or nearly quadratic depending on the property of the nonlinear function. Application is given to a model problem from chemical
engineering, and some numerical results, including a test problem with known analytical solution, are presented to illustrate
the various rates of convergence of the iterations.
Received November 2, 1995 / Revised version received February 10, 1997 相似文献
4.
A new superconvergence property of Wilson nonconforming finite element 总被引:13,自引:0,他引:13
Summary. In this paper the Wilson nonconforming finite element method is considered to solve a class of two-dimensional second-order
elliptic boundary value problems. A new superconvergence property at the vertices and the midpoints of four edges of rectangular
meshes is obtained.
Received May 5, 1995 / Revised version received November 11, 1996 相似文献
5.
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
6.
Summary.
We present a mixed finite element approximation
of an elliptic problem with degenerate coefficients, arising in the
study of the electromagnetic field in a resonant structure with
cylindrical symmetry. Optimal error bounds are derived.
Received
May 4, 1994 / Revised version received September 27, 1994 相似文献
7.
Summary. In this paper, we develop and analyze a new finite element method called the sparse finite element method for second order
elliptic problems. This method involves much fewer degrees of freedom than the standard finite element method. We show nevertheless
that such a sparse finite element method still possesses the superconvergence and other high accuracy properties same as those
of the standard finite element method. The main technique in our analysis is the use of some integral identities.
Received October 1, 1995 / Revised version received August 23, 1999 / Published online February 5, 2001 相似文献
8.
Summary.
Convergence for the spatial discretization by linear finite
elements of the non-parametric mean curvature flow is proved under natural
regularity assumptions on the continuous solution. Asymptotic convergence is
also obtained for the time derivative which is proportional to mean curvature.
An existence result for the continuous problem in adequate spaces is
included.
Received September 30, 1993 相似文献
9.
Superconvergence and a posteriori error estimation for
triangular mixed finite elements 总被引:5,自引:0,他引:5
Jan H. Brandts 《Numerische Mathematik》1994,68(3):311-324
Summary. In this paper,we prove superconvergence results for the vector
variable when lowest order triangular mixed finite elements of
Raviart-Thomas type [17] on uniform triangulations are used,
i.e., that the -distance between the
approximate solution and a suitable projection of the real solution
is of higher order than the -error. We
prove
results for both Dirichlet and Neumann boundary conditions. Recently,
Duran [9] proved similar results for rectangular mixed finite
elements, and superconvergence along the Gauss-lines for rectangular
mixed finite elements was considered by Douglas, Ewing, Lazarov and
Wang in [11], [8] and [18]. The triangular case
however needs some extra effort. Using the superconvergence results,
a simple postprocessing of the approximate solution will give an
asymptotically exact a posteriori error estimator for the
-error in the approximation of the vector variable.
Received December 6, 1992 / Revised
version received October 2, 1993 相似文献
10.
Summary. An unusual stabilized finite element is presented and analyzed herein for a generalized Stokes problem with a dominating
zeroth order term. The method consists in subtracting a mesh dependent term from the formulation without compromising consistency.
The design of this mesh dependent term, as well as the stabilization parameter involved, are suggested by bubble condensation.
Stability is proven for any combination of velocity and pressure spaces, under the hypotheses of continuity for the pressure
space. Optimal order error estimates are derived for the velocity and the pressure, using the standard norms for these unknowns.
Numerical experiments confirming these theoretical results, and comparisons with previous methods are presented.
Received April 26, 2001 / Revised version received July 30, 2001 / Published online October 17, 2001
Correspondence to: Gabriel R. Barrenechea 相似文献
11.
Summary. We consider the bilinear finite element approximation of smooth solutions to a simple parameter dependent elliptic model
problem, the problem of highly anisotropic heat conduction. We show that under favorable circumstances that depend on both
the finite element mesh and on the type of boundary conditions, the effect of parametric locking of the standard FEM can be
reduced by a simple variational crime. In our analysis we split the error in two orthogonal components, the approximation
error and the consistency error, and obtain different bounds for these separate components. Also some numerical results are
shown.
Received September 6, 1999 / Revised version received March 28, 2000 / Published online April 5, 2001 相似文献
12.
Summary. In this paper, we provide stability and convergence analysis for a class of finite difference schemes for unsteady incompressible
Navier-Stokes equations in vorticity-stream function formulation. The no-slip boundary condition for the velocity is converted
into local vorticity boundary conditions. Thom's formula, Wilkes' formula, or other local formulas in the earlier literature
can be used in the second order method; while high order formulas, such as Briley's formula, can be used in the fourth order
compact difference scheme proposed by E and Liu. The stability analysis of these long-stencil formulas cannot be directly
derived from straightforward manipulations since more than one interior point is involved in the formula. The main idea of
the stability analysis is to control local terms by global quantities via discrete elliptic regularity for stream function.
We choose to analyze the second order scheme with Wilkes' formula in detail. In this case, we can avoid the complicated technique
necessitated by the Strang-type high order expansions. As a consequence, our analysis results in almost optimal regularity
assumption for the exact solution. The above methodology is very general. We also give a detailed analysis for the fourth
order scheme using a 1-D Stokes model.
Received December 10, 1999 / Revised version received November 5, 2000 / Published online August 17, 2001 相似文献
13.
Summary. A monotone iterative method for numerical solutions of a class of finite difference reaction-diffusion equations with nonlinear
diffusion coefficient is presented. It is shown that by using an upper solution or a lower solution as the initial iteration
the corresponding sequence converges monotonically to a unique solution of the finite difference system. It is also shown
that the solution of the finite difference system converges to the solution of the continuous equation as the mesh size decreases
to zero.
Received February 18, 1998 / Revised version received April 21, 1999 / Published online February 17, 2000 相似文献
14.
Summary.
We consider the finite element approximation of a
non-Newtonian flow, where the viscosity obeys a general law including
the Carreau or power law. For sufficiently regular solutions we prove
energy type error bounds for the velocity and pressure. These bounds
improve on existing results in the literature. A key step in the
analysis is to prove abstract error bounds initially in a quasi-norm,
which naturally arises in degenerate problems of this type.
Received May 25, 1993 / Revised version received January 11, 1994 相似文献
15.
Tomás Chacón Rebollo 《Numerische Mathematik》1998,79(2):283-319
This paper introduces a stabilization technique for Finite Element numerical solution of 2D and 3D incompressible flow problems.
It may be applied to stabilize the discretization of the pressure gradient, and also of any individual operator term such
as the convection, curl or divergence operators, with specific levels of numerical diffusion for each one of them. Its computational
complexity is reduced with respect to usual (residual-based) stabilization techniques. We consider piecewise affine Finite
Elements, for which we obtain optimal error bounds for steady Navier-Stokes and also for generalized Stokes equations (including
convection). We include some numerical experiment in well known 2D test cases, that show its good performances.
Received March 15, 1996 / Revised version received January 17, 1997 相似文献
16.
Summary.
The aim of this work is to study a decoupled algorithm of
a fixed point for solving a
finite element (FE) problem for the approximation of viscoelastic
fluid flow obeying an Oldroyd B differential model. The interest for
this algorithm lies in its applications to numerical simulation and
in the cost of computing. Furthermore it is easy to bring this
algorithm into play.
The unknowns
are
the viscoelastic part of the extra stress tensor,
the velocity and
the pressure.
We suppose that the solution
is sufficiently
smooth and small. The approximation
of stress, velocity and pressure are resp.
discontinuous,
continuous,
continuous FE. Upwinding needed for convection of
, is made
by discontinuous FE. The method consists to
solve alternatively a transport equation for the stress,
and a Stokes like problem for velocity and pressure. Previously,
results of existence of the solution for the approximate problem and
error bounds have been obtained using fixed point
techniques with coupled algorithm.
In this paper we show that the mapping of the decoupled
fixed point algorithm is locally (in a neighbourhood of
)
contracting and we obtain existence, unicity (locally) of the solution
of the approximate problem and error bounds.
Received
July 29, 1994 / Revised version received March 13, 1995 相似文献
17.
Summary.
An adaptive finite element method for the calculation of transonic potential
flows was developed. An error indicator based on first order finite differences
of gradients is introduced as a local error estimator. It measures second order
distributional derivatives. Estimates involving
this error estimator, a residual and the error are given. The error estimator
can be used as a criterion for mesh refinement. We also give some computational
results.
Received September 16, 1993 / Revised version received June
7, 1994 相似文献
18.
Summary Robin interface conditions in domain decomposition methods enable the use of non overlapping subdomains and a speed up in
the convergence. Non conforming grids make the grid generation much easier and faster since it is then a parallel task. The
goal of this paper is to propose and analyze a new discretization scheme which allows to combine the use of Robin interface
conditions with non-matching grids. We consider both a symmetric definite positive operator and the convection-diffusion equation
discretized by finite volume schemes. Numerical results are shown.
Received December 22, 1999 / Revised version received December 21, 2000 / Published online December 18, 2001
Correspondence to: F. Nataf 相似文献
19.
The Mortar finite element method with Lagrange multipliers 总被引:19,自引:0,他引:19
Faker Ben Belgacem 《Numerische Mathematik》1999,84(2):173-197
Summary. The present paper deals with a variant of a non conforming domain decomposition technique: the mortar finite element method.
In the opposition to the original method this variant is never conforming because of the relaxation of the matching constraints at the vertices (and the edges in 3D) of subdomains. It
is shown that, written under primal hybrid formulation, the approximation problem, issued from a discretization of a second
order elliptic equation in 2D, is nonetheless well posed and provides a discrete solution that satisfies optimal error estimates
with respect to natural norms. Finally the parallelization advantages consequence of this variant are also addressed.
Received December 1, 1996 / Revised version received November 23, 1998 / Published online September 24, 1999 相似文献
20.
I. Perugia 《Numerische Mathematik》1999,84(2):305-326
Summary. A mixed field-based variational formulation for the solution of threedimensional magnetostatic problems is presented and
analyzed. This method is based upon the minimization of a functional related to the error in the constitutive magnetic relationship,
while constraints represented by Maxwell's equations are imposed by means of Lagrange multipliers. In this way, both the magnetic
field and the magnetic induction field can be approximated by using the most appropriate family of vector finite elements,
and boundary conditions can be imposed in a natural way. Moreover, this method is more suitable than classical approaches
for the approximation of problems featuring strong discontinuities of the magnetic permeability, as is usually the case. A
finite element discretization involving face and edge elements is also proposed, performing stability analysis and giving error estimates.
Received January 23, 1998 / Revised version received July 23, 1998 / Published online September 24, 1999 相似文献