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1.
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
2.
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
3.
A cascadic multigrid algorithm for the Stokes equations 总被引:4,自引:0,他引:4
A variant of multigrid schemes for the Stokes problem is discussed. In particular, we propose and analyse a cascadic version
for the Stokes problem. The analysis of the transfer between the grids requires special care in order to establish that the
complexity is the same as that for classical multigrid algorithms.
Received September 10, 1997 / Revised version received February 20, 1998 相似文献
4.
Summary. We examine the convergence characteristics of iterative methods based on a new preconditioning operator for solving the linear
systems arising from discretization and linearization of the steady-state Navier-Stokes equations. With a combination of analytic
and empirical results, we study the effects of fundamental parameters on convergence. We demonstrate that the preconditioned
problem has an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers. The
structure of these distributions is independent of the discretization mesh size, but the cardinality of the set of outliers
increases slowly as the viscosity becomes smaller. These characteristics are directly correlated with the convergence properties
of iterative solvers.
Received August 5, 2000 / Published online June 20, 2001 相似文献
5.
Summary.
The mortar element method is a
non conforming finite element method with
elements based on domain decomposition. For the Laplace equation,
it yields an ill conditioned linear system. For solving the linear system,
the so called preconditioned conjugate gradient method in
a subspace is used. Preconditioners are
proposed, and estimates on condition numbers
and arithmetical complexity are given.
Finally, numerical experiments are presented.
Received
June 22, 1994 / Revised version received February 6, 1995 相似文献
6.
A boundary multiplier/fictitious domain method for the steady incompressible Navier-Stokes equations
Summary. We analyze the error of a fictitious-domain method with boundary Lagrange multiplier. It is applied to solve a non-homogeneous steady incompressible Navier-Stokes problem in a domain with a multiply-connected boundary. The interior mesh in the fictitious domain and the boundary mesh are independent, up to a mesh-length ratio. Received February 24, 1999 / Revised version received January 30, 2000 / Published online October 16, 2000 相似文献
7.
Andrea Toselli 《Numerische Mathematik》2000,86(4):733-752
Summary. A two-level overlapping Schwarz method is considered for a Nédélec finite element approximation of 3D Maxwell's equations. For a fixed relative overlap, the condition number of the method is bounded, independently of the mesh size of the triangulation and the number of subregions. Our results are obtained with the assumption that the coarse triangulation is quasi-uniform and, for the Dirichlet problem, that the domain is convex. Our work generalizes well–known results for conforming finite elements for second order elliptic scalar equations. Numerical results for one and two-level algorithms are also presented. Received November 11, 1997 / Revised version received May 26, 1999 / Published online June 21, 2000 相似文献
8.
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
9.
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
10.
Multilevel diagonal scaling preconditioners for boundary element equations on locally refined meshes
Summary. We study a multilevel preconditioner for the Galerkin boundary element matrix arising from a symmetric positive-definite
bilinear form. The associated energy norm is assumed to be equivalent to a Sobolev norm of positive, possibly fractional,
order m on a bounded (open or closed) surface of dimension d, with . We consider piecewise linear approximation on triangular elements. Successive levels of the mesh are created by selectively
subdividing elements within local refinement zones. Hanging nodes may be created and the global mesh ratio can grow exponentially
with the number of levels. The coarse-grid correction consists of an exact solve, and the correction on each finer grid amounts
to a simple diagonal scaling involving only those degrees of freedom whose associated nodal basis functions overlap the refinement zone. Under appropriate assumptions on the choice of refinement zones, the condition number of the preconditioned system is shown
to be bounded by a constant independent of the number of degrees of freedom, the number of levels and the global mesh ratio.
In addition to applying to Galerkin discretisation of hypersingular boundary integral equations, the theory covers finite
element methods for positive-definite, self-adjoint elliptic problems with Dirichlet boundary conditions.
Received October 5, 2001 / Revised version received December 5, 2001 / Published online April 17, 2002
The support of this work through Visiting Fellowship grant GR/N21970 from the Engineering and Physical Sciences Research
Council of Great Britain is gratefully acknowledged. The second author was also supported by the Australian Research Council 相似文献
11.
Summary. We study some additive Schwarz algorithms for the version Galerkin boundary element method applied to some weakly singular and hypersingular integral equations of the first
kind. Both non-overlapping and overlapping methods are considered. We prove that the condition numbers of the additive Schwarz
operators grow at most as independently of h, where p is the degree of the polynomials used in the Galerkin boundary element schemes and h is the mesh size. Thus we show that additive Schwarz methods, which were originally designed for finite element discretisation
of differential equations, are also efficient preconditioners for some boundary integral operators, which are non-local operators.
Received June 15, 1997 / Revised version received July 7, 1998 / Published online February 17, 2000 相似文献
12.
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
13.
Susanne C. Brenner 《Numerische Mathematik》1996,72(4):419-447
Summary.
A two-level additive Schwarz preconditioner is
developed for the
systems resulting from the discretizations of
the plate bending problem by the Morley finite element, the
Fraeijs de Veubeke finite element, the Zienkiewicz finite
element and the Adini
finite element. The condition numbers of the preconditioned
systems are shown
to be bounded independent of mesh sizes and the number of
subdomains in the
case of a generous overlap.
Received
February 1, 1994 / Revised version received October 24, 1994 相似文献
14.
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
15.
Summary. Additive Schwarz preconditioners are developed for the p-version of the boundary element method for the hypersingular integral equation on surfaces in three dimensions. The principal
preconditioner consists of decomposing the subspace into local spaces associated with the element interiors supplemented with
a wirebasket space associated with the the element interfaces. The wirebasket correction involves inverting a diagonal matrix. If exact
solvers are used on the element interiors then theoretical analysis shows that growth of the condition number of the preconditioned
system is bounded by for an open surface and for a closed surface. A modified form of the preconditioner only requires the inversion of a diagonal matrix but results
in a further degradation of the condition number by a factor .
Received December 15, 1998 / Revised version received March 26, 1999 / Published online March 16, 2000 相似文献
16.
Multilevel Schwarz methods for elliptic problems
with discontinuous coefficients in three dimensions
Summary.
Multilevel Schwarz methods are developed for a
conforming finite element approximation of second order elliptic problems. We
focus on problems in three dimensions with
possibly large jumps in the coefficients across the
interface separating the subregions. We establish
a condition number estimate for the iterative operator, which is
independent of the coefficients, and grows at most as the square
of the number of levels. We also characterize a class of distributions
of the coefficients,
called quasi-monotone, for which the weighted
-projection is
stable and for which we can use the standard piecewise
linear functions as a coarse space. In this case,
we obtain optimal methods, i.e. bounds which are independent of the number
of levels and subregions. We also design and analyze multilevel
methods with new coarse spaces
given by simple explicit formulas. We consider nonuniform meshes
and conclude by an analysis of multilevel iterative substructuring methods.
Received April 6, 1994 / Revised version received December 7,
1994 相似文献
17.
Analysis of a pressure-stabilized finite element approximation of the stationary Navier-Stokes equations 总被引:1,自引:0,他引:1
Summary. The purpose of this paper is to analyze a finite element approximation of the stationary Navier-Stokes equations that allows the use of equal velocity-pressure interpolation. The idea is to introduce as unknown of the discrete problem the projection of the pressure gradient (multiplied by suitable algorithmic parameters) onto the space of continuous vector fields. The difference between these two vectors (pressure gradient and projection) is introduced in the continuity equation. The resulting formulation is shown to be stable and optimally convergent, both in a norm associated to the problem and in the norm for both velocities and pressure. This is proved first for the Stokes problem, and then it is extended to the nonlinear case. All the analysis relies on an inf-sup condition that is much weaker than for the standard Galerkin approximation, in spite of the fact that the present method is only a minor modification of this. Received May 25, 1998 / Revised version received August 31, 1999 / Published online July 12, 2000 相似文献
18.
Marcus Sarkis 《Numerische Mathematik》1997,77(3):383-406
Summary. Two-level domain decomposition methods are developed for a simple nonconforming approximation of second order elliptic problems.
A bound is established for the condition number of these iterative methods, that grows only logarithmically with the number
of degrees of freedom in each subregion. This bound holds for two and three dimensions and is independent of jumps in the
value of the coefficients and number of subregions. We introduce face coarse spaces, and isomorphisms to map between conforming
and nonconforming spaces.
ReceivedMarch 1, 1995 / Revised version received January 16, 1996 相似文献
19.
Gunther Schmidt 《Numerische Mathematik》1994,69(1):83-101
Summary.
The construction of a discretization of Poincar\'e--Steklov operators with the
boundary element method is given providing the same mapping properties as the
finite element discretization of these operators.
Received
October 1992 / Revised version received April 14, 1994 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