共查询到20条相似文献,搜索用时 15 毫秒
1.
In this paper we present a first supercloseness analysis for higher-order
Galerkin FEM applied to a singularly perturbed convection-diffusion problem. Using
a solution decomposition and a special representation of our finite element space,
we are able to prove a supercloseness property of $p + 1/4$ in the energy norm where
the polynomial order $p ≥ 3$ is odd. 相似文献
2.
In this paper,using Lin's integral identity technique,we prove the optimal uniform convergence θ(N_x~(-2)In~2N_x N_y~(-2)In~2N_y) in the L~2-norm for singularly per- turbed problems with parabolic layers.The error estimate is achieved by bilinear fi- nite elements on a Shishkin type mesh.Here N_x and N_y are the number of elements in the x- and y-directions,respectively.Numerical results are provided supporting our theoretical analysis. 相似文献
3.
研究了非协调有限元逼近非单调型拟线性椭圆问题,使用超收敛误差估计技巧,得出该问题光滑解和有限元解之间存在的超收敛关系. 相似文献
4.
We study the gradient superconvergence of bilinear finite volume
element (FVE) solving the elliptic problems. First, a superclose
weak estimate is established for the bilinear form of the FVE
method. Then, we prove that the gradient approximation of the FVE
solution has the superconvergence property: where denotes the average gradient on elements
containing point $P$ and $S$ is the set of optimal stress points
composed of the mesh points, the midpoints of edges and the centers of elements. 相似文献
5.
Sebastian Franz Torsten Linß 《Numerical Methods for Partial Differential Equations》2008,24(1):144-164
We analyze the superconvergence property of the Galerkin finite element method (FEM) for elliptic convection–diffusion problems with characteristic layers. This method on Shishkin meshes is known to be almost first‐order accurate (up to a logarithmic factor) in the energy norm induced by the bilinear form of the weak formulation, uniformly in the perturbation parameter. In the present paper the method is shown to be almost second‐order superconvergent in this energy norm for the difference between the FEM solution and the bilinear interpolant of the exact solution. This supercloseness property is used to improve the accuracy to almost second order by means of a postprocessing procedure. Numerical experiments confirm these results. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 相似文献
7.
Xiu Ye & Shangyou Zhang 《高等学校计算数学学报(英文版)》2021,14(3):613-623
This paper presents error estimates in both an energy norm and the $L^2$-norm for the weak Galerkin (WG) finite element methods for elliptic problems with
low regularity solutions. The error analysis for the continuous Galerkin finite element remains same regardless of regularity. A totally different analysis is needed for
discontinuous finite element methods if the elliptic regularity is lower than H-1.5.
Numerical results confirm the theoretical analysis. 相似文献
8.
In this paper, the discontinuous Galerkin method for the positive and symmetric, linear hyperbolic systems is constructed and analyzed by using bilinear finite elements on a rectangular domain, and an O(h2)-order superconvergence error estimate is established under the conditions of almost uniform partition and the H3-regularity for the exact solutions. The convergence analysis is based on some superclose estimates derived in this paper. Finally, as an application, the numerical treatment of Maxwell equation is discussed and computational results are presented. 相似文献
9.
对满足最大角条件和坐标系条件的二维区域中的各向异性一般三角形网格,研究了二阶椭圆问题的非协调Crouzeix-Raviart型线性三角形有限元逼近,得到了最优的能量模和L2-模误差估计结果. 相似文献
10.
In this paper, a new hybridized mixed formulation of weak Galerkin method is studied for a second order elliptic problem. This method is designed by approximate some operators with discontinuous piecewise polynomials in a shape regular finite element partition. Some discrete inequalities are presented on discontinuous spaces and optimal order error estimations are established. Some numerical results are reported to show super convergence and confirm the theory of the mixed weak Galerkin method. 相似文献
11.
Hans-Görg Roos 《Journal of Mathematical Analysis and Applications》2002,267(1):194-208
We consider a convection-diffusion problem with strong parabolic boundary layers and its discretization using upwind finite differences or bilinear finite elements on a layer-adapted mesh. Based on a new decomposition of the solution we are able to prove optimal uniform convergence results. 相似文献
12.
Yang Yang & Chi-Wang Shu 《计算数学(英文版)》2015,33(3):323-340
In this paper, we study the superconvergence of the error for the local discontinuous Galerkin (LDG) finite element method for one-dimensional linear parabolic equations when the alternating flux is used. We prove that if we apply piecewise $k$-th degree polynomials, the error between the LDG solution and the exact solution is ($k$+2)-th order superconvergent at the Radau points with suitable initial discretization. Moreover, we also prove the LDG solution is ($k$+2)-th order superconvergent for the error to a particular projection of the exact solution. Even though we only consider periodic boundary condition, this boundary condition is not essential, since we do not use Fourier analysis. Our analysis is valid for arbitrary regular meshes and for $P^k$ polynomials with arbitrary $k$ ≥ 1. We perform numerical experiments to demonstrate that the superconvergence rates proved in this paper are sharp. 相似文献
13.
Qun Lin & Rui-Feng Xie 《计算数学(英文版)》1989,7(4):402-411
In this paper, we derive the error expansion for finite element method under natural assumption and discuss the superconvergence as a special case of error expansion. 相似文献
14.
Che Sun 《计算数学(英文版)》1989,7(4):383-396
The Galerkin methods are studied for two-point boundary value problems and the related one-dimensional parabolic and hyperbolic problems. The boundary value problem considered here is of non-adjoint from and with mixed boundary conditions. The optimal order error estimate in the max-norm is first derived for the boundary problem for the finite element subspace. This result then gives optimal order max-norm error estimates for the continuous and discrete time approximations for the evolution problems described above. 相似文献
15.
In this paper, we will investigate the error estimates and the
superconvergence property of mixed finite element methods for a
semilinear elliptic control problem with an integral constraint on
control. The state and co-state are approximated by the lowest order
Raviart-Thomas mixed finite element and the control variable
is approximated by piecewise constant functions. We derive some
superconvergence properties for the control variable and the state
variables. Moreover, we derive $L^∞$- and $H^{-1}$-error
estimates both for the control variable and the state variables.
Finally, a numerical example is given to demonstrate the theoretical
results. 相似文献
16.
17.
The continuous Galerkin finite element method for linear delay-differential equation with several terms is studied. Adding some lower terms in the remainder of orthogonal expansion in an element so that the remainder satisfies more orthogonal condition in the element, and obtain a desired superclose function to finite element solution, thus the superconvergence of p -degree finite element approximate solution on (p+1)-order Lobatto points is derived. 相似文献
18.
A Comparison Study of Deep Galerkin Method and Deep Ritz Method for Elliptic Problems with Different Boundary Conditions 下载免费PDF全文
Recent years have witnessed growing interests in solving partial differential equations by deep neural networks, especially in the high-dimensional case. Unlike classical numerical methods, such as finite difference method and finite element method, the enforcement of boundary conditions in deep neural networks is highly nontrivial. One general strategy is to use the penalty method. In the work, we conduct a comparison study for elliptic problems with four different boundary conditions, i.e., Dirichlet, Neumann, Robin, and periodic boundary conditions, using two representative methods: deep Galerkin method and deep Ritz method. In the former, the PDE residual is minimized in the least-squares sense while the corresponding variational problem is minimized in the latter. Therefore, it is reasonably expected that deep Galerkin method works better for smooth solutions while deep Ritz method works better for low-regularity solutions. However, by a number of examples, we observe that deep Ritz method can outperform deep Galerkin method with a clear dependence of dimensionality even for smooth solutions and deep Galerkin method can also outperform deep Ritz method for low-regularity solutions.Besides, in some cases, when the boundary condition can be implemented in an exact manner, we find that such a strategy not only provides a better approximate solution but also facilitates the training process. 相似文献
19.
Lutz Angermann 《Numerical Methods for Partial Differential Equations》2002,18(2):241-259
This article investigates Petrov‐Galerkin discretizations of operator equations with linearly stable operators, where the residual does not belong to the annihilator W of the discrete test space Wh. Conforming and nonconforming methods are considered separately, and for the treatment of the nonconforming situation the concept of elliptic lifting is introduced. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 241–259, 2002; DOI 10.1002/num.1005 相似文献
20.
In this article, we investigate the superconvergence of the finite element approximation for optimal control problem governed by nonlinear elliptic equations. The state and co-state are discretized by piecewise linear functions and control is approximated by piecewise constant functions. We give the superconvergence analysis for both the control variable and the state variables. Finally, the numerical experiments show the theoretical results. 相似文献