共查询到20条相似文献,搜索用时 31 毫秒
1.
Hae‐Soo Oh Bongsoo Jang Yichung Jou 《Numerical Methods for Partial Differential Equations》2003,19(3):301-326
Recently Babus?ka‐Oh introduced the method of auxiliary mapping (MAM) which efficiently handles elliptic boundary value problems containing singularities. In this paper, a special weighted residue method, the Weighted Ritz‐Galerkin Method (WRGM), is investigated by introducing special weight functions. Together with this method, MAM is modified to yield highly accurate finite element solutions to general elliptic boundary value problems on the exterior of bounded domains at low cost. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 301–326, 2003. 相似文献
2.
Boniface Nkemzi Michael Jung 《Numerical Methods for Partial Differential Equations》2021,37(1):836-853
Solutions of boundary value problems in three‐dimensional domains with edges may exhibit singularities which are known to influence both the accuracy of the finite element solutions and the rate of convergence in the error estimates. This paper considers boundary value problems for the Poisson equation on typical domains Ω ? ?3 with edge singularities and presents, on the one hand, explicit computational formulas for the flux intensity functions. On the other hand, it proposes and analyzes a nonconforming finite element method on regular meshes for the efficient treatment of the singularities. The novelty of the present method is the use of the explicit formulas for the flux intensity functions in defining a postprocessing procedure in the finite element approximation of the solution. A priori error estimates in H1(Ω) show that the present algorithm exhibits the same rate of convergence as it is known for problems with regular solutions. 相似文献
3.
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
4.
Local mesh refinement for the discretization of Neumann boundary control problems on polyhedra 下载免费PDF全文
Thomas Apel Johannes Pfefferer Max Winkler 《Mathematical Methods in the Applied Sciences》2016,39(5):1206-1232
This paper deals with optimal control problems constrained by linear elliptic partial differential equations. The case where the right‐hand side of the Neumann boundary is controlled, is studied. The variational discretization concept for these problems is applied, and discretization error estimates are derived. On polyhedral domains, one has to deal with edge and corner singularities, which reduce the convergence rate of the discrete solutions, that is, one cannot expect convergence order two for linear finite elements on quasi‐uniform meshes in general. As a remedy, a local mesh refinement strategy is presented, and a priori bounds for the refinement parameters are derived such that convergence with optimal rate is guaranteed. As a by‐product, finite element error estimates in the H1(Ω)‐norm, L2(Ω)‐norm and L2(Γ)‐norm for the boundary value problem are obtained, where the latter one turned out to be the main challenge. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
5.
《Numerical Methods for Partial Differential Equations》2018,34(6):2079-2112
Highlights are the following:
- For any integer , we construct ‐continuous partition of unity (PU) functions with flat‐top from B‐spline functions to have numerical solutions of fourth‐order equations with singularities. B‐spline functions are modified to satisfy clamped boundary conditions.
- To handle singularity arising in fourth‐order elliptic differential equations, these modified B‐spline functions are enriched either by introducing enrichment basis functions implicitly through particular geometric mappings or by adding singular basis functions explicitly.
- To show the effectiveness of the proposed implicit enrichment methods (mapping method), the accuracy, the number of degrees of freedom (DOF), and matrix condition numbers are computed and compared in the h‐refinement, the p‐refinement, and the k‐refinement of the approximation space of B‐spline basis functions.
6.
Biyue Liu 《Numerical Methods for Partial Differential Equations》2003,19(2):152-166
In this article we present an analysis of a finite element method for solving two‐dimensional unsteady compressible Navier‐Stokes equations. Under the time‐stepping size restriction Δt ≤ Ch, we prove the existence and uniqueness of the numerical solution and obtain an a prior error estimate uniform in time. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 152–166, 2003 相似文献
7.
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
8.
Alexei Bespalov 《Numerical Methods for Partial Differential Equations》2008,24(4):1159-1180
The article considers a three‐dimensional crack problem in linear elasticity with Dirichlet boundary conditions. The crack in this model problem is assumed to be a smooth open surface with smooth boundary curve. The hp‐version of the boundary element method with weakly singular operator is applied to approximate the unknown jump of the traction which is not L2‐regular due to strong edge singularities. Assuming quasi‐uniform meshes and uniform distributions of polynomial degrees, we prove an a priori error estimate in the energy norm. The estimate gives an upper bound for the error in terms of the mesh size h and the polynomial degree p. It is optimal in h for any given data and quasi‐optimal in p for sufficiently smooth data. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008 相似文献
9.
Xindong Zhang Pengzhan Huang Xinlong Feng Leilei Wei 《Numerical Methods for Partial Differential Equations》2013,29(4):1081-1096
In this article, we consider the finite element method (FEM) for two‐dimensional linear time‐fractional Tricomi‐type equations, which is obtained from the standard two‐dimensional linear Tricomi‐type equation by replacing the first‐order time derivative with a fractional derivative (of order α, with 1 <α< 2 ). The method is based on finite element method for space and finite difference method for time. We prove that the method is unconditionally stable, and the error estimate is presented. The comparison of the FEM results with the exact solutions is made, and numerical experiments reveal that the FEM is very effective. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013 相似文献
10.
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
11.
Adaptive computation of smallest eigenvalues of self‐adjoint elliptic partial differential equations
We consider a new adaptive finite element (AFEM) algorithm for self‐adjoint elliptic PDE eigenvalue problems. In contrast to other approaches we incorporate the inexact solutions of the resulting finite‐dimensional algebraic eigenvalue problems into the adaptation process. In this way we can balance the costs of the adaptive refinement of the mesh with the costs for the iterative eigenvalue method. We present error estimates that incorporate the discretization errors, approximation errors in the eigenvalue solver and roundoff errors, and use these for the adaptation process. We show that it is also possible to restrict to very few iterations of a Krylov subspace solver for the eigenvalue problem on coarse meshes. Several examples are presented to show that this new approach achieves much better complexity than the previous AFEM approaches which assume that the algebraic eigenvalue problem is solved to full accuracy. Copyright © 2010 John Wiley & Sons, Ltd. 相似文献
12.
Shaohong Du Xiaoping Xie Pan Cheng 《Numerical Methods for Partial Differential Equations》2019,35(2):681-698
We consider a posteriori error estimation for a multipoint flux mixed finite element method for two‐dimensional elliptic interface problems. Within the class of modified quasi‐monotonically distributed coefficients, we derive a residual‐type a posteriori error estimator of the weighted sum of the scalar and flux errors which is robust with respect to the jumps of the coefficients. Moreover, we develop robust implicit and explicit recovery‐type estimators through gradient recovery in an H(curl)‐conforming finite element space. In particular, we apply a modified L2 projection in the implicit recovery procedure so as to reduce the computational cost of the recovered gradient. Numerical experiments confirm the theoretical results. 相似文献
13.
In this paper, Newmark time‐stepping scheme and edge elements are used to numerically solve the time‐dependent scattering problem in a three‐dimensional polyhedral cavity. Finite element methods based on the variational formulation derived in Van and Wood (Adv. Comput. Math., to appear) are considered. Existence and uniqueness of the discrete problem is proved by using Babuska–Brezzi theory. Finite element error estimate and stability of the Newmark scheme are also established. Copyright © 2003 John Wiley & Sons, Ltd. 相似文献
14.
Tongke Wang 《Numerical Methods for Partial Differential Equations》2008,24(1):24-40
On the basis of rectangular partition and bilinear interpolation, this article presents alternating direction finite volume element methods for two dimensional parabolic partial differential equations and gives three computational schemes, one is analogous to Douglas finite difference scheme with second order splitting error, the second has third order splitting error, and the third is an extended locally one dimensional scheme. Optimal L2 norm or H1 semi‐norm error estimates are obtained for these schemes. Finally, two numerical examples illustrate the effectiveness of the schemes. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 相似文献
15.
Guaranteed velocity error control for the pseudostress approximation of the Stokes equations 下载免费PDF全文
P. Bringmann C. Carstensen C. Merdon 《Numerical Methods for Partial Differential Equations》2016,32(5):1411-1432
The pseudostress approximation of the Stokes equations rewrites the stationary Stokes equations with pure (but possibly inhomogeneous) Dirichlet boundary conditions as another (equivalent) mixed scheme based on a stress in H(div) and the velocity in L2. Any standard mixed finite element function space can be utilized for this mixed formulation, e.g., the Raviart‐Thomas discretization which is related to the Crouzeix‐Raviart nonconforming finite element scheme in the lowest‐order case. The effective and guaranteed a posteriori error control for this nonconforming velocity‐oriented discretization can be generalized to the error control of some piecewise quadratic velocity approximation that is related to the discrete pseudostress. The analysis allows for local inf‐sup constants which can be chosen in a global partition to improve the estimation. Numerical examples provide strong evidence for an effective and guaranteed error control with very small overestimation factors even for domains with large anisotropy.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1411–1432, 2016 相似文献
16.
We develop a variational multiscale proper orthogonal decomposition (POD) reduced‐order model (ROM) for turbulent incompressible Navier‐Stokes equations. Under two assumptions on the underlying finite element approximation and the generation of the POD basis, the error analysis of the full discretization of the ROM is presented. All error contributions are considered: the spatial discretization error (due to the finite element discretization), the temporal discretization error (due to the backward Euler method), and the POD truncation error. Numerical tests for a three‐dimensional turbulent flow past a cylinder at Reynolds number show the improved physical accuracy of the new model over the standard Galerkin and mixing‐length POD ROMs. The high computational efficiency of the new model is also showcased. Finally, the theoretical error estimates are confirmed by numerical simulations of a two‐dimensional Navier‐Stokes problem. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 641–663, 2014 相似文献
17.
袁健华 《高等学校计算数学学报》2008,30(2)
1引言设Ω∈R~2为Lipschitz单连通的有界闭区域,X为定义在Ω的Sobolev空间,a(·,·)和b(·,·)为X×X→C的有界双线性或半双线性泛函,考虑变分特征值问题:求(λ,u≠0)∈C×X使得a(u,v)=λb(u,u),(?)u∈X,其中a(·,·)满足X上的"V-强制性"条件或者连续的inf-sup条件,设M_h为Q区域上的正则三角形剖分,X_h∈X为定义在M_h有限元子空间,上述变分问题对应的有限元离散问题为:求(λ_h,u_h)∈R×X,u_h≠0使得 相似文献
18.
Michael Karkulik Günther Of Dirk Praetorius 《Numerical Methods for Partial Differential Equations》2013,29(6):2081-2106
We consider the adaptive lowest‐order boundary element method based on isotropic mesh refinement for the weakly‐singular integral equation for the three‐dimensional Laplacian. The proposed scheme resolves both, possible singularities of the solution as well as of the given data. The implementation thus only deals with discrete integral operators, that is, matrices. We prove that the usual adaptive mesh‐refining algorithm drives the corresponding error estimator to zero. Under an appropriate saturation assumption which is observed empirically, the sequence of discrete solutions thus tends to the exact solution within the energy norm. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 相似文献
19.
Shin-Perng Chang 《Journal of Computational and Applied Mathematics》2011,235(13):3817-3824
This article concerns a procedure to generate optimal adaptive grids for convection dominated problems in two spatial dimensions based on least-squares finite element approximations. The procedure extends a one dimensional equidistribution principle which minimizes the interpolation error in some norms. The idea is to select two directions which can reflect the physics of the problems and then apply the one dimensional equidistribution principle to the chosen directions. Model problems considered are the two dimensional convection-diffusion problems where boundary and interior layers occur. Numerical results of model problems illustrating the efficiency of the proposed scheme are presented. In addition, to avoid skewed mesh in the optimal grids generated by the algorithm, an unstructured local mesh smoothing will be considered in the least-squares approximations. Comparisons with the Gakerkin finite element method will also be provided. 相似文献
20.
In this study, numerical simulations of the improved Boussinesq equation are obtained using two finite difference schemes and two finite element methods, based on the second‐and third‐order time discretization. The methods are tested on the problems of propagation of a soliton and interaction of two solitons. After the L∞ error norm is used to measure differences between the exact and numerical solutions, the results obtained by the proposed methods are compared with recently published results. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010 相似文献