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1.
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
2.
Finite volume element methods for non-definite problems 总被引:8,自引:0,他引:8
Ilya D. Mishev 《Numerische Mathematik》1999,83(1):161-175
Summary. The error estimates for finite volume element method applied to 2 and 3-D non-definite problems are derived. A simple upwind scheme is proven to be unconditionally stable and first order accurate. Received August 27, 1997 / Revised version received May 12, 1998 相似文献
3.
Least-squares mixed finite element methods
for non-selfadjoint elliptic problems: I. Error estimates
Summary.
A least-squares mixed finite element
method for general second-order non-selfadjoint
elliptic problems in two- and three-dimensional domains
is formulated and analyzed. The finite element spaces for
the primary solution approximation
and the flux approximation
consist of piecewise polynomials of degree
and respectively.
The method is mildly nonconforming on the boundary.
The cases and
are studied.
It is proved that the method is not subject to the LBB-condition.
Optimal - and
-error estimates are derived for
regular finite element partitions.
Numerical experiments, confirming the theoretical rates of
convergence, are presented.
Received
October 15, 1993 / Revised version received August 2, 1994 相似文献
4.
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
5.
Summary. In this paper we study the numerical behaviour of elliptic
problems in which a small parameter is involved and an example
concerning the computation of elastic arches is analyzed using this
mathematical framework. At first, the statements of the problem and its
Galerkin approximations are defined and an asymptotic
analysis is performed. Then we give general conditions ensuring that
a numerical scheme will converge uniformly with respect to the small
parameter. Finally we study an example in
computation of arches working in linear elasticity conditions. We build one
finite element scheme giving a locking behaviour, and another one
which does not.
Revised version received October 25, 1993 相似文献
6.
Summary. We develop the general a priori error analysis of residual-free bubble finite element approximations to non-self-adjoint elliptic problems of the form subject to homogeneous Dirichlet boundary condition, where A is a symmetric second-order elliptic operator, C is a skew-symmetric first-order differential operator, and is a positive parameter. Optimal-order error bounds are derived in various norms, using piecewise polynomial finite elements of degree . Received October 1, 1998/ Revised version received April 6, 1999 / Published online January 27, 2000 相似文献
7.
Summary. The use of mixed finite element methods is well-established in the numerical approximation of the problem of nearly incompressible elasticity, and its limit, Stokes flow. The question of stability over curved elements for such methods is of particular significance in the p version, where, since the element size remains fixed, exact representation of the curved boundary by (large) elements is often used. We identify a mixed element which we show to be optimally stable in both p and h refinement over curvilinear meshes. We prove optimal p version (up to ) and h version (p = 2, 3) convergence for our element, and illustrate its optimality through numerical experiments. Received August 25, 1998 / Revised version received February 16, 1999 / Published online April 20, 2000 –? Springer-Verlag 2000 相似文献
8.
Nonlinear Galerkin methods and mixed finite elements:
two-grid algorithms for the Navier-Stokes equations 总被引:14,自引:0,他引:14
Summary.
A nonlinear Galerkin method using mixed finite
elements is presented for the two-dimensional
incompressible Navier-Stokes equations. The
scheme is based on two finite element spaces
and for the approximation of the velocity,
defined respectively on one coarse grid with grid
size and one fine grid with grid size and
one finite element space for the approximation
of the pressure. Nonlinearity and time
dependence are both treated on the coarse space.
We prove that the difference between the new
nonlinear Galerkin method and the standard
Galerkin solution is of the order of $H^2$, both in
velocity ( and pressure norm).
We also discuss a penalized version of our algorithm
which enjoys similar properties.
Received October 5, 1993 / Revised version received November
29, 1993 相似文献
9.
Summary. We consider the bidimensional Stokes problem for incompressible fluids in stream function-vorticity. For this problem, the
classical finite element method of degree one converges only in for the norm of the vorticity. We propose to use harmonic functions to approach the vorticity along the boundary. Discrete harmonics
are functions that are used in practice to derive a new numerical method. We prove that we obtain with this numerical scheme
an error of order for the norm of the vorticity.
Received January, 2000 / Revised version received May 15, 2001 / Published online December 18, 2001 相似文献
10.
Summary. This analysis of convergence of a coupled FEM-IEM is based on our previous work on the FEM and the IEM for exterior Helmholtz
problems. The key idea is to represent both the exact and the numerical solution by the Dirichlet-to-Neumann operators that
they induce on the coupling hypersurface in the exterior of an obstacle. The investigation of convergence can then be related
to a spectral analysis of these DtN operators. We give a general outline of our method and then proceed to a detailed investigation
of the case that the coupling surface is a sphere. Our main goal is to explore the convergence mechanism. In this context,
we show well-posedness of both the continuous and the discrete models. We further show that the discrete inf-sup constants
have a positive lower bound that does not depend on the number of DOF of the IEM. The proofs are based on lemmas on the spectra
of the continuous and the discrete DtN operators, where the spectral characterization of the discrete DtN operator is given
as a conjecture from numerical experiments. In our convergence analysis, we show algebraic (in terms of N) convergence of arbitrary order and generalize this result to exponential convergence.
Received April 10, 1999 / Revised version received November 10, 1999 / Published online October 16, 2000 相似文献
11.
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
12.
Summary. Multilevel preconditioners are proposed for the iterative solution of the discrete problems which arise when orthogonal spline
collocation with piecewise Hermite bicubics is applied to the Dirichlet boundary value problem for a self-adjoint elliptic
partial differential equation on a rectangle. Additive and multiplicative preconditioners are defined respectively as sums
and products of independent operators on a sequence of nested subspaces of the fine partition approximation space. A general
theory of additive and multiplicative Schwarz methods is used to prove that the preconditioners are spectrally equivalent
to the collocation discretization of the Laplacian with the spectral constants independent of the fine partition stepsize
and the number of levels. The preconditioned conjugate gradient and preconditioned Orthomin methods are considered for the
solution of collocation problems. An implementation of the methods is discussed and the results of numerical experiments are
presented.
Received March 1, 1994 / Revised version received January 23, 1996 相似文献
13.
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
14.
Finite element methods and their convergence for elliptic and parabolic interface problems 总被引:5,自引:0,他引:5
In this paper, we consider the finite element methods for solving second order elliptic and parabolic interface problems
in two-dimensional convex polygonal domains. Nearly the same optimal -norm and energy-norm error estimates as for regular problems are obtained when the interfaces are of arbitrary shape but
are smooth, though the regularities of the solutions are low on the whole domain. The assumptions on the finite element triangulation
are reasonable and practical.
Received July 7, 1996 / Revised version received March 3, 1997 相似文献
15.
Summary. In this paper, we study a multiscale finite element method for solving a class of elliptic problems with finite number of
well separated scales. The method is designed to efficiently capture the large scale behavior of the solution without resolving
all small scale features. This is accomplished by constructing the multiscale finite element base functions that are adaptive
to the local property of the differential operator. The construction of the base functions is fully decoupled from element
to element; thus the method is perfectly parallel and is naturally adapted to massively parallel computers. We present the
convergence analysis of the method along with the results of our numerical experiments. Some generalizations of the multiscale
finite element method are also discussed.
Received April 17, 1998 / Revised version received March 25, 2000 / Published online June 7, 2001 相似文献
16.
To the best knowledge of the authors, this work presents the first convergence analysis for the Infinite Element Method (IEM)
for the Helmholtz equation in exterior domains. The approximation applies to separable geometries only, combining an arbitrary
Finite Element (FE) discretization on the boundary of the domain with a spectral-like approximation in the “radial” direction,
with shape functions resulting from the separation of variables. The principal idea of the presented analysis is based on
the spectral decomposition of the problem.
Received February 10, 1996 / Revised version received February 17, 1997 相似文献
17.
Summary. The Schur complement of a model problem is considered as a preconditioner for the Uzawa type schemes for the generalized
Stokes problem (the Stokes problem with the additional term in the motion equation). The implementation of the preconditioned method requires for each iteration only one extra solution
of the Poisson equation with Neumann boundary conditions. For a wide class of 2D and 3D domains a theorem on its convergence
is proved. In particular, it is established that the method converges with a rate that is bounded by some constant independent
of . Some finite difference and finite element methods are discussed. Numerical results for finite difference MAC scheme are
provided.
Received May 2, 1997 / Revised version received May 10, 1999 / Published online May 8, 2000 相似文献
18.
Alexander Ženíšek 《Numerische Mathematik》1995,71(3):399-417
Summary.
The finite element method for an elliptic equation with discontinuous
coefficients (obtained for the magnetic potential from Maxwell's
equations) is analyzed in the union of closed domains the boundaries
of which form a system of three circles with the same centre.
As the middle domain is very narrow the triangulations obeying
the maximum angle condition are considered. In the case of piecewise
linear trial functions the maximum rate of
convergence in the norm
of the space is proved
under the following conditions:
1. the exact solution
is piecewise of class ;
2. the family of subtriangulations
of the narrow
subdomain satisfies the maximum angle condition
expressed by relation (38). The paper extends the results of [24].
Received
March 8, 1993 / Revised version received November 28, 1994 相似文献
19.
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
20.
Summary. We consider a second-order elliptic equation with discontinuous or anisotropic coefficients in a bounded two- or three dimensional domain, and its finite-element discretization. The aim of this paper is to prove some a priori and a posteriori error estimates in an appropriate norm, which are independent of the variation of the coefficients. Received February 5, 1999 / Published online March 16, 2000 相似文献