共查询到20条相似文献,搜索用时 515 毫秒
1.
Danxia Wang Yanan Li Hongen Jia 《Numerical Methods for Partial Differential Equations》2023,39(2):1251-1265
In this paper, we present a two-grid finite element method for the Allen-Cahn equation with the logarithmic potential. This method consists of two steps. In the first step, based on a fully implicit finite element method, the Allen-Cahn equation is solved on a coarse grid with mesh size H. In the second step, a linearized system whose nonlinear term is replaced by the value of the first step is solved on a fine grid with mesh size h. We give the energy stabilities of the traditional finite element method and the two-grid finite element method. The optimal convergence order of the two-grid finite element method in H1 norm is achieved when the mesh sizes satisfy h = O(H2). Numerical examples are given to demonstrate the validity of the proposed scheme. The results show that the two-grid method can save the CPU time while keeping the same convergence rate. 相似文献
2.
Norbert Heuer 《Numerische Mathematik》2001,88(3):485-511
Summary. We analyze an additive Schwarz preconditioner for the p-version of the boundary element method for the single layer potential operator on a plane screen in the three-dimensional
Euclidean space. We decompose the ansatz space, which consists of piecewise polynomials of degree p on a mesh of size h, by introducing a coarse mesh of size . After subtraction of the coarse subspace of piecewise constant functions on the coarse mesh this results in local subspaces
of piecewise polynomials living only on elements of size H. This decomposition yields a preconditioner which bounds the spectral condition number of the stiffness matrix by . Numerical results supporting the theory are presented.
Received August 15, 1998 / Revised version received November 11, 1999 / Published online December 19, 2000 相似文献
3.
Chunjia Bi 《Numerical Methods for Partial Differential Equations》2007,23(1):220-233
In this article, we consider the finite volume element method for the second‐order nonlinear elliptic problem and obtain the H1 and W1,∞ superconvergence estimates between the solution of the finite volume element method and that of the finite element method, which reveal that the finite volume element method is in close relationship with the finite element method. With these superconvergence estimates, we establish the Lp and W1,p (2 < p ≤ ∞) error estimates for the finite volume element method for the second‐order nonlinear elliptic problem. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 相似文献
4.
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
5.
Sven Beuchler 《Numerical Linear Algebra with Applications》2003,10(8):721-732
From the literature it is known that the conjugate gradient method with domain decomposition preconditioners is one of the most efficient methods for solving systems of linear algebraic equations resulting from p‐version finite element discretizations of elliptic boundary value problems. One ingredient of such a preconditioner is a preconditioner related to the Dirichlet problems. In the case of Poisson's equation, we present a preconditioner for the Dirichlet problems which can be interpreted as the stiffness matrix Kh,k resulting from the h‐version finite element discretization of a special degenerated problem. We construct an AMLI preconditioner Ch,k for the matrix Kh,k and show that the condition number of C Kh,k is independent of the discretization parameter. This proof is based on the strengthened Cauchy inequality. The theoretical result is confirmed by numerical examples. Copyright © 2003 John Wiley & Sons, Ltd. 相似文献
6.
An additive Schwarz preconditioner for nonconforming mortar finite element discretization of a second order elliptic problem
in two dimensions with arbitrary large jumps of the discontinuous coefficients in subdomains is described.
An almost optimal estimate of the condition number of the preconditioned problem is proved. The number of preconditioned conjugate
gradient iterations is independent of jumps of the coefficients and is proportional to (1+log(H/h)), where H,h are mesh sizes.
AMS subject classification (2000) 65N55, 65N30, 65N22 相似文献
7.
Norikazu Saito Yoshiki Sugitani 《Numerical Methods for Partial Differential Equations》2019,35(1):181-199
Convergence results are presented for the immersed boundary (IB) method applied to a model Stokes problem. As a discretization method, we use the finite element method. First, the immersed force field is approximated using a regularized delta function. Its error in the W?1, p norm is examined for 1 ≤ p < n/(n ? 1), with n representing the space dimension. Subsequently, we consider IB discretization of the Stokes problem and examine the regularization and discretization errors separately. Consequently, error estimate of order h1 ? α in the W1, 1 × L1 norm for the velocity and pressure is derived, where α is an arbitrary small positive number. The validity of those theoretical results is confirmed from numerical examples. 相似文献
8.
YING Long'an 《中国科学A辑(英文版)》2000,43(9):945-957
A second order explicit finite element scheme is given for the numerical computation to multi-dimensional scalar conservation
laws.L
p
convergence to entropy solutions is proved under some usual conditions. For two-dimensional problems, uniform mesh, and sufficiently
smooth solutions a second order error estimate inL
2 is proved under a stronger condition, Δt≤Ch
2/4 相似文献
9.
We treat the finite volume element method (FVE) for solving general second order elliptic problems as a perturbation of the linear finite element method (FEM), and obtain the optimal H1 error estimate, H1 superconvergence and Lp (1 < p ≤ ∞) error estimates between the solution of the FVE and that of the FEM. In particular, the superconvergence result does not require any extra assumptions on the mesh except quasi‐uniform. Thus the error estimates of the FVE can be derived by the standard error estimates of the FEM. Moreover we consider the effects of numerical integration and prove that the use of barycenter quadrature rule does not decrease the convergence orders of the FVE. The results of this article reveal that the FVE is in close relationship with the FEM. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 693–708, 2003. 相似文献
10.
Comparison of multigrid algorithms for high‐order continuous finite element discretizations 下载免费PDF全文
Hari Sundar Georg Stadler George Biros 《Numerical Linear Algebra with Applications》2015,22(4):664-680
We present a comparison of different multigrid approaches for the solution of systems arising from high‐order continuous finite element discretizations of elliptic partial differential equations on complex geometries. We consider the pointwise Jacobi, the Chebyshev‐accelerated Jacobi, and the symmetric successive over‐relaxation smoothers, as well as elementwise block Jacobi smoothing. Three approaches for the multigrid hierarchy are compared: (1) high‐order h‐multigrid, which uses high‐order interpolation and restriction between geometrically coarsened meshes; (2) p‐multigrid, in which the polynomial order is reduced while the mesh remains unchanged, and the interpolation and restriction incorporate the different‐order basis functions; and (3) a first‐order approximation multigrid preconditioner constructed using the nodes of the high‐order discretization. This latter approach is often combined with algebraic multigrid for the low‐order operator and is attractive for high‐order discretizations on unstructured meshes, where geometric coarsening is difficult. Based on a simple performance model, we compare the computational cost of the different approaches. Using scalar test problems in two and three dimensions with constant and varying coefficients, we compare the performance of the different multigrid approaches for polynomial orders up to 16. Overall, both h‐multigrid and p‐multigrid work well; the first‐order approximation is less efficient. For constant coefficients, all smoothers work well. For variable coefficients, Chebyshev and symmetric successive over‐relaxation smoothing outperform Jacobi smoothing. While all of the tested methods converge in a mesh‐independent number of iterations, none of them behaves completely independent of the polynomial order. When multigrid is used as a preconditioner in a Krylov method, the iteration number decreases significantly compared with using multigrid as a solver. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
11.
We present a preconditioner for the linearized Navier–Stokes equations which is based on the combination of a fast transform approximation of an advection diffusion problem together with the recently introduced ‘BFBTT’ preconditioner of Elman (SIAM Journal of Scientific Computing, 1999; 20 :1299–1316). The resulting preconditioner when combined with an appropriate Krylov subspace iteration method yields the solution in a number of iterations which appears to be independent of the Reynolds number provided a mesh Péclet number restriction holds, and depends only mildly on the mesh size. The preconditioner is particularly appropriate for problems involving a primary flow direction. Copyright © 2001 John Wiley & Sons, Ltd. 相似文献
12.
Mario A. Casarin 《Numerische Mathematik》2001,89(2):307-339
Summary. The - spectral element discretization of the Stokes equation gives rise to an ill-conditioned, indefinite, symmetric linear system
for the velocity and pressure degrees of freedom. We propose a domain decomposition method which involves the solution of
a low-order global, and several local problems, related to the vertices, edges, and interiors of the subdomains. The original
system is reduced to a symmetric equation for the velocity, which can be solved with the conjugate gradient method. We prove
that the condition number of the iteration operator is bounded from above by , where C is a positive constant independent of the degree N and the number of subdomains, and is the inf-sup condition of the pair -. We also consider the stationary Navier-Stokes equations; in each Newton step, a non-symmetric indefinite problem is solved
using a Schwarz preconditioner. By using an especially designed low-order global space, and the solution of local problems
analogous to those decribed above for the Stokes equation, we are able to present a complete theory for the method. We prove
that the number of iterations of the GMRES method, at each Newton step, is bounded from above by . The constant C does not depend on the number of subdomains or N, and it does not deteriorate as the Newton iteration proceeds.
Received March 2, 1998 / Revised version received October 12, 1999 / Published online March 20, 2001 相似文献
13.
Chunmei Wang 《Applications of Mathematics》2014,59(6):653-672
In this paper, we consider mortar-type Crouzeix-Raviart element discretizations for second order elliptic problems with discontinuous coefficients. A preconditioner for the FETI-DP method is proposed. We prove that the condition number of the preconditioned operator is bounded by (1 + log(H/h))2, where H and h are mesh sizes. Finally, numerical tests are presented to verify the theoretical results. 相似文献
14.
We propose an almost optimal preconditioner for the iterative solution of the Galerkin equations arising from a hypersingular integral equation on an interval. This preconditioning technique, which is based on the single layer potential, was already studied for closed curves [11,14]. For a boundary element trial space, we show that the condition number is of order (1 + | log h
min|)2, where h
min is the length of the smallest element. The proof requires only a mild assumption on the mesh, easily satisfied by adaptive refinement algorithms. 相似文献
15.
Bedřich Sousedík Roger G. Ghanem Eric T. Phipps 《Numerical Linear Algebra with Applications》2014,21(1):136-151
Use of the stochastic Galerkin finite element methods leads to large systems of linear equations obtained by the discretization of tensor product solution spaces along their spatial and stochastic dimensions. These systems are typically solved iteratively by a Krylov subspace method. We propose a preconditioner, which takes an advantage of the recursive hierarchy in the structure of the global matrices. In particular, the matrices posses a recursive hierarchical two‐by‐two structure, with one of the submatrices block diagonal. Each of the diagonal blocks in this submatrix is closely related to the deterministic mean‐value problem, and the action of its inverse is in the implementation approximated by inner loops of Krylov iterations. Thus, our hierarchical Schur complement preconditioner combines, on each level in the approximation of the hierarchical structure of the global matrix, the idea of Schur complement with loops for a number of mutually independent inner Krylov iterations, and several matrix–vector multiplications for the off‐diagonal blocks. Neither the global matrix nor the matrix of the preconditioner need to be formed explicitly. The ingredients include only the number of stiffness matrices from the truncated Karhunen–Loève expansion and a good preconditioned for the mean‐value deterministic problem. We provide a condition number bound for a model elliptic problem, and the performance of the method is illustrated by numerical experiments. Copyright © 2013 John Wiley & Sons, Ltd. 相似文献
16.
So‐Hsiang Chou Do Y. Kwak Qian Li 《Numerical Methods for Partial Differential Equations》2003,19(4):463-486
We consider convergence of the covolume or finite volume element solution to linear elliptic and parabolic problems. Error estimates and superconvergence results in the Lp norm, 2 ≤ p ≤ ∞, are derived. We also show second‐order convergence in the Lp norm between the covolume and the corresponding finite element solutions and between their gradients. The main tools used in this article are an extension of the “supercloseness” results in Chou and Li [Math Comp 69(229) (2000), 103–120] to the Lp based spaces, duality arguments, and the discrete Green's function method. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 463–486, 2003 相似文献
17.
We consider a new preconditioning technique for the iterative solution of linear systems of equations that arise when discretizing
partial differential equations. The method is applied to finite difference discretizations, but the ideas apply to other discretizations
too.
If E is a fundamental solution of a differential operator P, we have E*(Pu) = u. Inspired by this, we choose the preconditioner to be a discretization of an approximate inverse K, given by a convolution-like operator with E as a kernel.
We present analysis showing that if P is a first order differential operator, KP is bounded, and numerical results show grid independent convergence for first order partial differential equations, using
fixed point iterations.
For the second order convection-diffusion equation convergence is no longer grid independent when using fixed point iterations,
a result that is consistent with our theory. However, if the grid is chosen to give a fixed number of grid points within boundary
layers, the number of iterations is independent of the physical viscosity parameter.
AMS subject classification (2000) 65F10, 65N22 相似文献
18.
Discontinuous Galerkin finite element methods for variational inequalities of first and second kinds
J.K. Djoko 《Numerical Methods for Partial Differential Equations》2008,24(1):296-311
We develop the error analysis for the h‐version of the discontinuous Galerkin finite element discretization for variational inequalities of first and second kinds. We establish an a priori error estimate for the method which is of optimal order in a mesh dependant as well as L2‐norm.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007 相似文献
19.
Ivar Gustafsson 《BIT Numerical Mathematics》1996,36(1):86-100
It is well known that standard incomplete factorization (IC) methods exist for M-matrices [15] and that modified incomplete factorization (MIC) methods exist for weakly diagonally dominant matrices [8]. The restriction to these classes of matrices excludes many realistic general applications to discretized partial differential equations. We present a technique to avoid this problem by making an initial modification already at the element level, followed by the standard IC or MIC factorization of the assembled matrix. This modification ensures weakly diagonally dominant M-matrices and is made in such a way that the condition number of the matrix is only increased by a constant factor independent of the mesh parameterh. Hence the fast convergence of the MICCG method, that is inO(h
–1/2),h 0 iterations for second order elliptic problems, is preserved. 相似文献
20.
S. Battal Gazi Karakoc Michael Neilan 《Numerical Methods for Partial Differential Equations》2014,30(4):1254-1278
A symmetric C 0 finite element method for the biharmonic problem is constructed and analyzed. In our approach, we introduce one‐sided discrete second‐order derivatives and Hessian matrices to formulate our scheme. We show that the method is stable and converge with optimal order in a variety of norms. A distinctive feature of the method is that the results hold without extrinsic penalization of the gradient across interelement boundaries. Numerical experiments are given that support the theoretical results, and the extension to Kirchhoff plates is also discussed. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1254–1278, 2014 相似文献