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1.
Anisotropic mesh refinement
in stabilized Galerkin methods 总被引:8,自引:0,他引:8
Summary.
The numerical solution of a convection-diffusion-reaction model problem is
considered in two and three dimensions. A stabilized finite element method
of Galerkin/Least-square type accomodates diffusion-dominated as well as
convection- and/or reaction-dominated situations. The resolution of
boundary layers occuring in the singularly perturbed case is achieved
using anisotropic mesh refinement in boundary layer regions.
In this paper, the
standard analysis of the stabilized Galerkin method on isotropic meshes
is extended to more general meshes with boundary layer refinement.
Simplicial Lagrangian elements of arbitrary order are used.
Received
March 6, 1995 / Revised version received August 18,
1995 相似文献
2.
Jürgen Bey 《Numerische Mathematik》2000,85(1):1-29
Summary. In the present paper we investigate Freudenthal's simplex refinement algorithm which can be considered to be the canonical
generalization of Bank's well known red refinement strategy for triangles. Freudenthal's algorithm subdivides any given (n)-simplex into subsimplices, in such a way that recursive application results in a stable hierarchy of consistent triangulations. Our investigations
concentrate in particular on the number of congruence classes generated by recursive refinements. After presentation of the
method and the basic ideas behind it, we will show that Freudenthal's algorithm produces at most n!/2 congruence classes for any initial (n)-simplex, no matter how many subsequent refinements are performed. Moreover, we will show that this number is optimal in
the sense that recursive application of any affine invariant refinement strategy with sons per element results in at least
n!/2 congruence classes for almost all (n)-simplices.
Received February 23, 1998/ Revised version received December 9, 1998 / Published online January 27, 2000 相似文献
3.
Kunibert G. Siebert 《Numerische Mathematik》1996,73(3):373-398
Summary.
Besides an algorithm for local refinement, an a posteriori error
estimator is the basic tool of every adaptive finite element
method. Using information generated by such an error estimator the
refinement of the grid is controlled. For 2nd order elliptic
problems we present an error estimator for anisotropically refined
grids, like -d cuboidal and 3-d prismatic grids, that gives
correct information about the size of the error; additionally it
generates information about the direction into which some element
has to be refined to reduce the error in a proper way. Numerical
examples are presented for 2-d rectangular and 3-d prismatic grids.
Received March 15, 1994 / Revised version received June 3, 1994 相似文献
4.
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
5.
Summary.
This paper presents general algorithms for the parallel
solution of
finite element problems associated with maximal monotone operators of local
type. The latter concept, which is also introduced here, is well suited to
capture the idea that the given operator is the discretization of a differential
operator that may involve nonlinearities and/or constraints as long as those are
of a local nature. Our algorithms are obtained as a combination of known
algorithms for possibly multi-valued maximal monotone operators with appropriate
decompositions of the domain. This work extends a method due to two of the
authors in the single-valued and linear case.
Received April 25, 1994 相似文献
6.
Summary. We consider the isoparametric transformation, which maps a given reference element onto a global element given by its vertices,
for multi-linear finite elements on pyramids and prisms. We present easily computable conditions on the position of the vertices,
which ensure that the isoparametric transformation is bijective.
Received May 7, 1999 / Revised version received April 28, 2000 / Published online December 19, 2000 相似文献
7.
When trains of impulse controls are present on the right-hand side of a system of ordinary differential equations, the solution
is no longer smooth and contains jumps which can accumulate at several points in the time interval. In technological and physical
systems the sum of the absolute value of all the impulses is finite and hence the total variation of the solution is finite.
So the solution at best belongs to the space BV of vector functions with bounded variation. Unless variable node methods are
used, the loss of smoothness of the solution would a priori make higher-order methods over a fixed mesh inactractive. Indeed
in general the order of -convergence is and the nodal rate is . However in the linear case -convergence rate remains but the nodal rate can go up to by using one-step or multistep scheme with a nodal rate up to when the solution belongs to . Proofs are given of error estimates and several numerical experiments confirm the optimality of the estimates.
Received March 15, 1996 / Revised version received January 3, 1997 相似文献
8.
Summary. Macro-elements of arbitrary smoothness are constructed on Clough-Tocher triangle splits. These elements can be used for solving
boundary-value problems or for interpolation of Hermite data, and are shown to be optimal with respect to spline degree. We
conjecture they are also optimal with respect to the number of degrees of freedom. The construction provides local bases for
certain superspline spaces defined over Clough-Tocher refinements of arbitrary triangulations. These bases are shown to be
stable as a function of the smallest angle in the triangulation, which in turn implies that the associated spline spaces have
optimal order approximation power.
Received November 18, 1999 / Published online October 16, 2000 相似文献
9.
Summary. The aim of this paper is to prove some Babuška–Brezzi type conditions which are involved in the mortar spectral element discretization
of the Stokes problem, for several cases of nonconforming domain decompositions.
ID=" <E5>Dedicated to Olof B. Widlund on the occasion of his 60th birthday</E5> 相似文献
10.
Superconvergence analysis and error expansion for the Wilson nonconforming finite element 总被引:8,自引:0,他引:8
Summary.
In this paper the Wilson nonconforming finite element is considered for
solving a class of two-dimensional second-order elliptic boundary value
problems. Superconvergence estimates and error expansions are obtained
for both uniform and non-uniform rectangular meshes. A new lower bound
of the error shows that the usual error estimates are optimal. Finally
a discussion on the error behaviour in negative norms shows that there
is generally no improvement in the order by going to weaker norms.
Received July 5, 1993 相似文献
11.
Summary. Three iterative domain decomposition methods are considered: simultaneous updates on all subdomains (Additive Schwarz Method),
flow directed sweeps and double sweeps. By using some techniques of formal language theory we obtain a unique criterion
of convergence for the three methods. The convergence rate is a function of the criterion and depends on the algorithm.
Received October 24, 1994 / Revised version received November 27, 1995 相似文献
12.
This paper deals with a posteriori estimates for the finite element solution of the Stokes problem in stream function and vorticity formulation. For two different
discretizations, we propose error indicators and we prove estimates in order to compare them with the local error. In a second
step, these results are extended to the Navier-Stokes equations.
Received March 25, 1996 / Revised version received April 7, 1997 相似文献
13.
Summary. A unified approach to construct finite elements based on a dual-hybrid formulation of the linear elasticity problem is given.
In this formulation the stress tensor is considered but its symmetry is relaxed by a Lagrange multiplier which is nothing
else than the rotation. This construction is linked to the approximations of the Stokes problem in the primitive variables
and it leads to a new interpretation of known elements and to new finite elements. Moreover all estimates are valid uniformly
with respect to compressibility and apply in the incompressible case which is close to the Stokes problem.
Received June 20, 1994 / Revised version received February 16, 1996 相似文献
14.
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
15.
Summary. We present an adaptive finite element method for solving elliptic problems in exterior domains, that is for problems in the
exterior of a bounded closed domain in , . We describe a procedure to generate a sequence of bounded computational domains , , more precisely, a sequence of successively finer and larger grids, until the desired accuracy of the solution is reached. To this end we prove an a posteriori error estimate for the error on the unbounded domain in the energy norm
by means of a residual based error estimator. Furthermore we prove convergence of the adaptive algorithm. Numerical examples
show the optimal order of convergence.
Received July 8, 1997 /Revised version received October 23, 1997 相似文献
16.
A mean-field model for superconductivity is studied from both the analytical and computational points of view. In this model,
the individual vortex-like structures occuring in practical superconductors are not resolved. Rather, these structures are
homogenized and a vortex density is solved for. The particular model studied includes effects due to the pinning of vortices.
The existence and uniqueness of solutions of a regularized version of the model are demonstrated and the behavior of these
solutions as the regularization parameter tends to zero is examined. Then, semi-discrete and fully discrete finite element
based discretizations are formulated and analyzed and the results of some computational experiments are presented.
Received January 21, 1997 相似文献
17.
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
18.
Patrick Lacoste 《Numerische Mathematik》2000,84(4):577-609
Summary. This study deals with the mathematical and numerical solution of time-harmonic Maxwell equation in axisymmetric geometry.
Using Fourier decomposition, we define weighted Sobolev spaces of solution and we prove expected regularity results. A practical
contribution of this paper is the construction of a class of finite element conforming with the H (rot) space equipped with the weighted measure rdrdz. It appears as an extension of the well-known cartesian mixed finite element of Raviart-Thomas-Nédélec [11]–[15]. These elements
are built from classical lagrangian and mixed finite element, therefore no special approximations functions are needed. Finally,
following works of Mercier and Raugel [10], we perform an interpolation error estimate for the simplest proposed element.
Received March 15, 1996 / Revised version received November 30, 1998 / Published online December 6, 1999 相似文献
19.
Mark Ainsworth 《Numerische Mathematik》1996,73(4):399-418
Summary.
The element residual method for a posteriori error estimation is analyzed
for degree finite element approximation on quadrilateral elements.
The influence of the choice of subspace used to solve the element residual
problem is studied. It is shown that the resulting estimators will be
consistent (or asymptotically exact) for all
if and only if
the mesh is parallel. Moreover, even if the mesh consists of rectangles, then
the estimators can be inconsistent when .
The results provide concrete guidelines for the selection of a posteriori
error estimators and establish the limits of their performance. In particular,
the use of the element residual method for high orders of approximation
(such as those arising in the - version finite element method) is
vindicated.
The mechanism behind the rather poor performance of the estimators is traced
back to the basic formulation of the residual problem. The investigations
reveal a deficiency in the formulation, leading, as it does, to spurious
modes in the true solution of the residual problem. The recommended choice
of subspaces may be viewed as being sufficient to guarantee that the spurious
modes are filtered out from the approximate solution while at the same time
retaining a sufficient degree of approximation to represent the true modes.
Received February 27, 1995 / Revised version
received June 7, 1995 相似文献
20.
Summary. The saturation assumption asserts that the best approximation error in with piecewise quadratic finite elements is strictly smaller than that of piecewise linear finite elements. We establish
a link between this assumption and the oscillation of , and prove that small oscillation relative to the best error with piecewise linears implies the saturation assumption. We
also show that this condition is necessary, and asymptotically valid provided .
Received November 17, 2000 / Published online July 25, 2001 相似文献