共查询到20条相似文献,搜索用时 15 毫秒
1.
Koichi Niijima 《Numerische Mathematik》1989,56(7):707-719
Summary Pointwise error estimates for a streamline diffusion scheme for solving a model convection-dominated singularly perturbed convection-diffusion problem are given. These estimates improve pointwise error estimates obtained by Johnson et al.[5]. 相似文献
2.
Martin Stynes 《Milan Journal of Mathematics》1994,64(1):129-140
We consider a model time-dependent convection-diffusion problem, whose solution may exhibit interior and boundary layers. The standard streamline diffusion scheme, with piecewise linear elements on a uniform mesh, will converge only at points that are not close to any layer. We replace the uniform mesh by a special piecewise uniform mesh that is chosen a priori and resolves part of any outflow boundary layer. The resulting method is convergent, independently of the diffusion parameter, with a pointwise accuracy of almost order 5/4 away from layers and almost order 3/4 inside the boundary layer. 相似文献
3.
A model singularly perturbed convection–diffusion problem in two space dimensions is considered. The problem is solved by a streamline diffusion finite element method (SDFEM) that uses piecewise bilinear finite elements on a Shishkin mesh. We prove that the method is convergent, independently of the diffusion parameter ε, with a pointwise accuracy of almost order 11/8 outside and inside the boundary layers. Numerical experiments support these theoretical results. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 相似文献
4.
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
5.
Finite element superconvergence on Shishkin mesh for 2-D convection-diffusion problems 总被引:4,自引:0,他引:4
Zhimin Zhang. 《Mathematics of Computation》2003,72(243):1147-1177
In this work, the bilinear finite element method on a Shishkin mesh for convection-diffusion problems is analyzed in the two-dimensional setting. A superconvergence rate in a discrete -weighted energy norm is established under certain regularity assumptions. This convergence rate is uniformly valid with respect to the singular perturbation parameter . Numerical tests indicate that the rate is sharp for the boundary layer terms. As a by-product, an -uniform convergence of the same order is obtained for the -norm. Furthermore, under the same regularity assumption, an -uniform convergence of order in the norm is proved for some mesh points in the boundary layer region.
6.
《Journal of Computational and Applied Mathematics》1987,19(3):389-393
Pointwise error estimates are obtained for polynomial interpolants in the roots and extrema of the Chebyshev polynomials of the first kind. These estimates are analogous to those derived by Henrici [2] for trigonometric polynomial interpolants. 相似文献
7.
In this paper, we present pointwise estimates of the streamline diffusion finite element method (SDFEM) for conforming piecewise linears on Shishkin triangular meshes. The method is applied to a model singularly perturbed convection-diffusion problem with characteristic layers. Using a new variant of artificial crosswind diffusion, we prove that uniformly pointwise error bounds away from the layers are of order almost 7/4 (up to a logarithmic factor). In some cases, the convergence order is almost 15/8. Our analysis depends on discrete Green’s functions and sharp estimates of the diffusion and convection parts in the bilinear form. Finally, numerical experiments support our theoretical results. 相似文献
8.
In this paper, a linearized semi-implicit finite difference scheme is proposed to solve the strongly coupled fractional Ginzburg-Landau equations. The difference scheme, which involves three time levels, is unconditionally stable, fourth-order accurate in space, and second-order accurate in time. By using the energy method and mathematical induction, the unique solvability, the unconditional stability, and optimal pointwise error estimate are obtained. Finally, some numerical experiments are presented to validate our theoretical findings. 相似文献
9.
Alejandro Allendes Gilberto Campaa Erwin Hernndez 《Mathematical Methods in the Applied Sciences》2019,42(10):3549-3567
We derive pointwise error estimates for a generalized Oseen when it is approximated by a low order Taylor‐Hood finite element scheme in two dimensions. The analysis is based on estimates for regularized Green's functions associated with a generalized Oseen problem on weighted Sobolev spaces and weighted interpolation results. We apply the maximum norm results to obtain convergence in an optimal control problem governed by a generalized Oseen equation and present a numerical example that allows us to show the behavior of the error. 相似文献
10.
Ricardo H. Nochetto Alfred Schmidt Kunibert G. Siebert Andreas Veeser 《Numerische Mathematik》2006,104(4):515-538
We derive upper and lower a posteriori estimates for the maximum norm error in finite element solutions of monotone semi-linear equations. The estimates hold for Lagrange elements of any fixed order, non-smooth nonlinearities, and take numerical integration into account. The proof hinges on constructing continuous barrier functions by correcting the discrete solution appropriately, and then applying the continuous maximum principle; no geometric mesh constraints are thus required. Numerical experiments illustrate reliability and efficiency properties of the corresponding estimators and investigate the performance of the resulting adaptive algorithms in terms of the polynomial order and quadrature. 相似文献
11.
In this article, we analyze the local superconvergence property of the streamline-diffusion finite-element method (SDFEM) for scalar convection-diffusion problems with dominant convection. By orienting the mesh in the streamline direction and imposing a uniformity condition on the mesh, the theoretical order of pointwise convergence is increased from O(h11/8|log h|) to O(h2|log h|). Numerical tests show that this result cannot be extended to arbitrary quasi-uniform meshes. © 1996 John Wiley & Sons, Inc. 相似文献
12.
Hongsen Chen. 《Mathematics of Computation》2005,74(251):1097-1116
In this paper we derive some pointwise error estimates for the local discontinuous Galerkin (LDG) method for solving second-order elliptic problems in (). Our results show that the pointwise errors of both the vector and scalar approximations of the LDG method are of the same order as those obtained in the norm except for a logarithmic factor when the piecewise linear functions are used in the finite element spaces. Moreover, due to the weighted norms in the bounds, these pointwise error estimates indicate that when at least piecewise quadratic polynomials are used in the finite element spaces, the errors at any point depend very weakly on the true solution and its derivatives in the regions far away from . These localized error estimates are similar to those obtained for the standard conforming finite element method.
13.
Peter Henrici 《Journal of Computational and Applied Mathematics》1982,8(2):131-132
An estimate due to Gaier [2] for the error committed in replacing a periodic function f by an interpolating trigonometric polynomial is sharpened in such a way that the estimate makes evident the interpolating property of the polynomial. A similar improvement is given for Gaier's estimate of the difference between the conjugate of f and the conjugate trigonometric polynomial. 相似文献
14.
Summary. We consider singularly perturbed linear elliptic problems in two dimensions. The solutions of such problems typically exhibit layers and are difficult to solve numerically. The streamline diffusion finite element method (SDFEM) has been proved to produce accurate solutions away from any layers on uniform meshes, but fails to compute the boundary layers precisely. Our modified SDFEM is implemented with piecewise linear functions on a Shishkin mesh that resolves boundary layers, and we prove that it yields an accurate approximation of the solution both inside and outside these layers. The analysis is complicated by the severe nonuniformity of the mesh. We give local error estimates that hold true uniformly in the perturbation parameter , provided only that , where mesh points are used. Numerical experiments support these theoretical results. Received February 19, 1999 / Revised version received January 27, 2000 / Published online August 2, 2000 相似文献
15.
16.
Jaeun Ku 《BIT Numerical Mathematics》2010,50(3):609-630
Highly localized pointwise error estimates for a stabilized Galerkin method are provided for second-order non-selfadjoint
elliptic partial differential equations. The estimates show a local dependence of the error on the derivative of the solution
u and weak dependence on the global norm. The results in this paper are an extension of the previous pointwise error estimates
for the self-adjoint problems. In order to provide pointwise error estimates in the presence of the first-order term in the
differential equations, we prove that the stabilized Galerkin solution is higher order perturbation to the Ritz projection
of the true solutions. Then, we proceed to obtain pointwise estimates using the so-called discrete Green’s function. Application
to error expansion inequalities and a posteriori error estimators are briefly discussed. 相似文献
17.
Parallel Galerkin domain decomposition procedures based on the streamline diffusion method for convection-diffusion problems 总被引:1,自引:0,他引:1
Based upon the streamline diffusion method, parallel Galerkin domain decomposition procedures for convection-diffusion problems are given. These procedures use implicit method in the sub-domains and simple explicit flux calculations on the inter-boundaries of sub-domains by integral mean method or extrapolation method to predict the inner-boundary conditions. Thus, the parallelism can be achieved by these procedures. The explicit nature of the flux calculations induces a time step limitation that is necessary to preserve stability. Artificial diffusion parameters δ are given. By analysis, optimal order error estimate is derived in a norm which is stronger than L2-norm for these procedures. This error estimate not only includes the optimal H1-norm error estimate, but also includes the error estimate along the streamline direction ‖β⋅∇(u−U)‖, which cannot be achieved by standard finite element method. Experimental results are presented to confirm theoretical results. 相似文献
18.
Summary. We derive pointwise weighted error estimates for a semidiscrete finite element method applied to parabolic equations. The results extend those obtained by A.H. Schatz for stationary elliptic problems. In particular, they show that the error is more localized for higher order elements.
Mathematics Subject Classification (2000): 65N30 相似文献
19.
G. P. Galdi B. Straughan 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》1982,33(3):334-343
Summary Pointwise estimates are obtained for the perturbation velocity to a novel theory of fluid jet behaviour based on Cosserat continuum mechanics. The method employed is one of weighted energy.
Sommario In questo lavoro si determinano stime puntuali per il campo perturbato di velocità relativo ad una nuova teoria di getto fluido. Tale teoria è basata sul modello della meccanica dei continui di tipo Cosserat. Il metodo impiegato è quello dell'energia pesata.相似文献
20.
《Journal of Computational and Applied Mathematics》2005,182(2):372-387
In this paper a singularly perturbed Riccati initial value problem is examined. Parameter explicit bounds on the solution and its derivatives are given. A numerical method composed of an implicit difference operator and a piecewise-uniform Shishkin mesh is constructed. A theoretical parameter independently bound on the errors in the numerical approximations is established. Numerical results are presented which are in agreement with the theoretical error bound. 相似文献