共查询到20条相似文献,搜索用时 15 毫秒
1.
Christos Xenophontos Scott R. Fulton 《Numerical Methods for Partial Differential Equations》2003,19(1):89-111
We consider the numerical approximation of singularly perturbed reaction‐diffusion problems over two‐dimensional domains with smooth boundary. Using the h version of the finite element method over appropriately designed piecewise uniform (Shishkin) meshes, we are able to uniformly approximate the solution at a quasi‐optimal rate. The results of numerical computations showing agreement with the analysis are also presented. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 89–111, 2003 相似文献
2.
Zhimin Zhang 《Numerical Methods for Partial Differential Equations》2002,18(3):374-395
Superconvergence approximations of singularly perturbed two‐point boundary value problems of reaction‐diffusion type and convection‐diffusion type are studied. By applying the standard finite element method of any fixed order p on a modified Shishkin mesh, superconvergence error bounds of (N?1 ln (N + 1))p+1 in a discrete energy norm in approximating problems with the exponential type boundary layers are established. The error bounds are uniformly valid with respect to the singular perturbation parameter. Numerical tests indicate that the error estimates are sharp; in particular, the logarithmic factor is not removable. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 374–395, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10001 相似文献
3.
Gunar Matthies 《Advances in Computational Mathematics》2009,30(4):315-337
We consider a singularly perturbed convection-diffusion equation on the unit square where the solution of the problem exhibits
exponential boundary layers. In order to stabilise the discretisation, two techniques are combined: Shishkin meshes are used
and the local projection method is applied. For arbitrary r≥2, the standard Q
r
-element is enriched by just six additional functions leading to an element which contains the P
r+1. In the local projection norm, the difference between the solution of the stabilised discrete problem and an interpolant
of the exact solution is of order uniformly in ε. Furthermore, it is shown that the method converges uniformly in ε of order in the global energy norm.
相似文献
4.
A uniformly convergent continuous-discontinuous Galerkin method for singularly perturbed problems of convection-diffusion type 总被引:1,自引:0,他引:1
In this paper, we introduce a coupled approach of local discontinuous Galerkin and standard finite element method for solving singularly perturbed convection-diffusion problems. On Shishkin mesh with linear elements, a rate O(N-1lnN) in an associated norm is established, where N is the number of elements. Numerical experiments complement the theoretical results. Moreover, a rate O(N-2ln2N) in a discrete L∞ norm, and O(N-2) in L2 norm, are observed numerically on the Shishkin mesh. 相似文献
5.
Torsten Linß 《Numerical Methods for Partial Differential Equations》2000,16(5):426-440
We consider a Galerkin finite element method that uses piecewise bilinears on a class of Shishkin‐type meshes for a model singularly perturbed convection‐diffusion problem on the unit square. The method is shown to be convergent, uniformly in the diffusion parameter ϵ, of almost second order in a discrete weighted energy norm. As a corollary, we derive global L2‐norm error estimates and local L∞‐norm estimates. Numerical experiments support our theoretical results. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16:426–440, 2000 相似文献
6.
A uniformly convergent numerical method for a coupled system of two singularly perturbed linear reaction-diffusion problems 总被引:2,自引:0,他引:2
A coupled system of two singularly perturbed linear reactiondiffusiontwo-point boundary value problems is examined. The leading termof each equation is multiplied by a small positive parameter,but these parameters may have different magnitudes. The solutionsto the system have boundary layers that overlap and interact.The structure of these layers is analysed, and this leads tothe construction of a piecewise-uniform mesh that is a variantof the usual Shishkin mesh. On this mesh central differencingis proved to be almost first-order accurate, uniformly in bothsmall parameters. Supporting numerical results are presentedfor a test problem. 相似文献
7.
A mixed hp FEM for the approximation of fourth‐order singularly perturbed problems on smooth domains
P. Constantinou S. Franz L. Ludwig C. Xenophontos 《Numerical Methods for Partial Differential Equations》2019,35(1):114-127
We consider fourth‐order singularly perturbed problems posed on smooth domains and the approximation of their solution by a mixed Finite Element Method on the so‐called Spectral Boundary Layer Mesh. We show that the method converges uniformly, with respect to the singular perturbation parameter, at an exponential rate when the error is measured in the energy norm. Numerical examples illustrate our theoretical findings. 相似文献
8.
Considering a two‐dimensional singularly perturbed convection–diffusion problem with exponential boundary layers, we analyze the local discontinuous Galerkin (DG) method that uses piecewise bilinear polynomials on Shishkin mesh. A convergence rate O(N‐1 lnN) in a DG‐norm is established under the regularity assumptions, while the total number of mesh points is O(N2). The rate of convergence is uniformly valid with respect to the singular perturbation parameter ε. Numerical experiments indicate that the theoretical error estimate is sharp. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013 相似文献
9.
Kaushik Mukherjee Srinivasan Natesan 《Numerical Methods for Partial Differential Equations》2014,30(6):1931-1960
In this article, we consider a class of singularly perturbed mixed parabolic‐elliptic problems whose solutions possess both boundary and interior layers. To solve these problems, a hybrid numerical scheme is proposed and it is constituted on a special rectangular mesh which consists of a layer resolving piecewise‐uniform Shishkin mesh in the spatial direction and a uniform mesh in the temporal direction. The domain under consideration is partitioned into two subdomains. For the spatial discretization, the proposed scheme is comprised of the classical central difference scheme in the first subdomain and a hybrid finite difference scheme in the second subdomain, whereas the time derivative in the given problem is discretized by the backward‐Euler method. We prove that the method converges uniformly with respect to the perturbation parameter with almost second‐order spatial accuracy in the discrete supremum norm. Numerical results are finally presented to validate the theoretical results.© 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1931–1960, 2014 相似文献
10.
In this paper, a new DG method was designed to solve the model problem of the one-dimensional singularly-perturbed convection-diffusion equation. With some special chosen numerical traces, the existence and uniqueness of the DG solution is provided. The superconvergent points inside each element are observed. Particularly, the 2p + 1-order superconvergence and even uniform superconvergence under layer-adapted mesh are observed numerically. 相似文献
11.
The nonconforming cell boundary element (CBE) methods are proposed. The methods are designed in such a way that they enjoy the mass conservation at the element level and the normal component of fluxes at inter-element boundaries are continuous for unstructured triangular meshes. Normal flux continuity and the optimal order error estimates in a broken H1 norm for the P1 method are established, which are completion of authors' earlier works [Y. Jeon, D. Sheen, Analysis of a cell boundary element method, Adv. Comput. Math. 22 (3) (2005) 201–222; Y. Jeon, E.-J. Park, D. Sheen, A cell boundary element method for elliptic problems, Numer. Methods Partial Differential Equations 21 (3) (2005) 496–511]. Moreover, two second order methods (the and modified methods) and a multiscale CBE method are constructed and numerical experiments are performed. Numerical results show feasibility and effectiveness of the CBE methods. 相似文献
12.
The combination technique for a two-dimensional convection-diffusion problem with exponential layers
Sebastian Franz Fang Liu Hans-Görg Roos Martin Stynes Aihui Zhou 《Applications of Mathematics》2009,54(3):203-223
Convection-diffusion problems posed on the unit square and with solutions displaying exponential layers are solved using a
sparse grid Galerkin finite element method with Shishkin meshes. Writing N for the maximum number of mesh intervals in each
coordinate direction, our “combination” method simply adds or subtracts solutions that have been computed by the Galerkin
FEM on and meshes. It is shown that the combination FEM yields (up to a factor ln N) the same order of accuracy in the associated energy norm as the Galerkin FEM on an N × N mesh, but it requires only (N
3/2) degrees of freedom compared with the (N
2) used by the Galerkin FEM. An analogous result is also proved for the streamline diffusion finite element method.
This work was supported by the National Natural Science Foundation of China (10701083 and 10425105), the Chinese National
Basic Research Program (2005CB321704) and the Boole Centre for Research in Informatics at National University of Ireland Cork. 相似文献
13.
We consider the numerical approximation of singularly perturbed elliptic boundary value problems over nonsmooth domains. We use a decomposition of the solution that contains a smooth part, a corner layer part and a boundary layer part. Explicit guidelines for choosing mesh‐degree combinations are given that yield finite element spaces with robust approximation properties. In particular, we construct an hp finite element space that approximates all components uniformly, at a near exponential rate. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 63–89, 1999 相似文献
14.
In this paper, we construct a kind of novel finite difference (NFD) method for solving singularly perturbed reaction–diffusion problems. Different from directly truncating the high‐order derivative terms of the Taylor's series in the traditional finite difference method, we rearrange the Taylor's expansion in a more elaborate way based on the original equation to develop the NFD scheme for 1D problems. It is proved that this approach not only can highly improve the calculation accuracy but also is uniformly convergent. Then, applying alternating direction implicit technique, the newly deduced schemes are extended to 2D equations, and the uniform error estimation based on Shishkin mesh is derived, too. Finally, numerical experiments are presented to verify the high computational accuracy and theoretical prediction. 相似文献
15.
We consider streamline diffusion finite element methods applied to a singularly perturbed convection–diffusion two‐point boundary
value problem whose solution has a single boundary layer. To analyse the convergence of these methods, we rewrite them as
finite difference schemes. We first consider arbitrary meshes, then, in analysing the scheme on a Shishkin mesh, we consider
two formulations on the fine part of the mesh: the usual streamline diffusion upwinding and the standard Galerkin method.
The error estimates are given in the discrete L
∞ norm; in particular we give the first analysis that shows precisely how the error depends on the user-chosen parameter τ0 specifying the mesh. When τ0 is too small, the error becomes O(1), but for τ0 above a certain threshold value, the error is small and increases either linearly or quadratically as a function of . Numerical
tests support our theoretical results.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
16.
In this article we transform a large class of parabolic inverse problems into a nonclassical parabolic equation whose coefficients
consist of trace type functionals of the solution and its derivatives subject to some initial and boundary conditions. For
this nonclassical problem, we study finite element methods and present an immediate analysis for global superconvergence for
these problems, on basis of which we obtain a posteriori error estimators.
This research was supported in part by the Shahid Beheshti University, the National Basic Research Program of China (2007CB814906),
the National Natural Science Foundation of China (10471103 and 10771158), Social Science Foundation of the Ministry of Education
of China (Numerical methods for convertible bonds, 06JA630047), Tianjin Natural Science Foundation (07JCYBJC14300). 相似文献
17.
Hans‐Görg Roos Helena Zarin 《Numerical Methods for Partial Differential Equations》2007,23(6):1560-1576
We consider a convection–diffusion problem with Dirichlet boundary conditions posed on a unit square. The problem is discretized using a combination of the standard Galerkin FEM and an h–version of the nonsymmetric discontinuous Galerkin FEM with interior penalties on a layer–adapted mesh with linear/bilinear elements. With specially chosen penalty parameters for edges from the coarse part of the mesh, we prove uniform convergence (in the perturbation parameter) in an associated norm. In the same norm we also establish a supercloseness result. Numerical tests support our theoretical estimates.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 相似文献
18.
19.
We study the uniform approximation of boundary layer functions for , , by the and versions of the finite element method. For the version (with fixed mesh), we prove super-exponential convergence in the range . We also establish, for this version, an overall convergence rate of in the energy norm error which is uniform in , and show that this rate is sharp (up to the term) when robust estimates uniform in are considered. For the version with variable mesh (i.e., the version), we show that exponential convergence, uniform in , is achieved by taking the first element at the boundary layer to be of size . Numerical experiments for a model elliptic singular perturbation problem show good agreement with our convergence estimates, even when few degrees of freedom are used and when is as small as, e.g., . They also illustrate the superiority of the approach over other methods, including a low-order version with optimal ``exponential" mesh refinement. The estimates established in this paper are also applicable in the context of corresponding spectral element methods.
20.
Long Chen Yonggang Wang Jinbiao Wu 《Journal of Computational and Applied Mathematics》2008,220(1-2):712-724
A one-dimensional singularly perturbed problem with a boundary turning point is considered in this paper. Let Vh be the linear finite element space on a suitable grid . A variant of streamline diffusion finite element method is proved to be almost uniform stable in the sense that the numerical approximation uh satisfies u-uh∞C|lnε| infvhVhu-vh∞, where C is independent with the small diffusion coefficient ε and the mesh . Such stability result is applied to layer-adapted grids to obtain almost ε-uniform second order scheme for turning point problems. 相似文献