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1.
Natural superconvergence of the least-squares finite element method is surveyed for the one-and two-dimensional Poisson equation.
For two-dimensional problems, both the families of Lagrange elements and Raviart-Thomas elements have been considered on uniform
triangular and rectangular meshes. Numerical experiments reveal that many superconvergence properties of the standard Galerkin
method are preserved by the least-squares finite element method.
The second author was supported in part by the US National Science Foundation under Grant DMS-0612908. 相似文献
2.
Madhusmita Tripathy 《Applicable analysis》2013,92(8):1213-1231
In this article, we study the semidiscrete H 1-Galerkin mixed finite element method for parabolic problems over rectangular partitions. The well-known optimal order error estimate in the L 2-norm for the flux is of order 𝒪(h k+1) (SIAM J. Numer. Anal. 35 (2), (1998), pp. 712–727), where k ≥ 1 is the order of the approximating polynomials employed in the Raviart–Thomas element. We derive a superconvergence estimate of order 𝒪(h k+3) between the H 1-Galerkin mixed finite element approximation and an appropriately defined local projection of the flux variable when k ≥ 1. A the new approximate solution for the flux with superconvergence of order 𝒪(h k+3) is realized via a postprocessing technique using local projection methods. 相似文献
3.
Bo Li 《Numerical Methods for Partial Differential Equations》2004,20(1):33-59
We consider the finite element approximation of the Laplacian operator with the homogeneous Dirichlet boundary condition, and study the corresponding Lagrange interpolation in the context of finite element superconvergence. For d‐dimensional Qk‐type elements with d ≥ 1 and k ≥ 1, we prove that the interpolation points must be the Lobatto points if the Lagrange interpolation and the finite element solution are superclose in H1 norm. For d‐dimensional Pk‐type elements, we consider the standard Lagrange interpolation—the Lagrange interpolation with interpolation points being the principle lattice points of simplicial elements. We prove for d ≥ 2 and k ≥ d + 1 that such interpolation and the finite element solution are not superclose in both H1 and L2 norms and that not all such interpolation points are superconvergence points for the finite element approximation. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 33–59, 2004. 相似文献
4.
A. A. Panin 《Computational Mathematics and Mathematical Physics》2008,48(12):2211-2216
The coincidence of an approximate solution to the boundary value problem for an ordinary differential equation with the exact solution at mesh nodes is proved for a certain class of the generalized finite element methods. 相似文献
5.
1.IntroductionLetfibeaplanedomainwithsmoothboundaryonandWm,p(fl)betheusualSobolevspaceonnwithnormWhenp=2,pisusuallyomitted.WeshalldenotetheusualinnerproductinL'(fl)orLa(O)'by','),andinL'(ofl)by't').Weshallusethesamenotationstoindicatethedualltiesbetw... 相似文献
6.
Jan H. Brandts 《Applications of Mathematics》2009,54(3):225-235
We show that a non-standard mixed finite element method proposed by Barrios and Gatica in 2007, is a higher order perturbation
of the least-squares mixed finite element method. Therefore, it is also superconvergent whenever the least-squares mixed finite
element method is superconvergent. Superconvergence of the latter was earlier investigated by Brandts, Chen and Yang between
2004 and 2006. Since the new method leads to a non-symmetric system matrix, its application seems however more expensive than
applying the least-squares mixed finite element method.
Dedicated to Ivan Hlaváček on the occasion of his 75th birthday 相似文献
7.
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
8.
In this paper, the global superconvergence is analysed on two schemes (a mixed finite element scheme and a finite element scheme) for Maxwell's equations in . Such a supercovergence analysis is achieved by means of the technique of integral identity (which has been used in the supercovergence analysis for many other equations and schemes) on a rectangular mesh, and then are generalized into more general domains and problems with the variable coefficients. Besides being more direct, our analysis generalizes the results of Monk.
9.
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
10.
Qiding Zhu 《Applications of Mathematics》1998,43(6):401-411
In 1995, Wahbin presented a method for superconvergence analysis called Interior symmetric method, and declared that it is universal. In this paper, we carefully examine two superconvergence techniques used by mathematicians both in China and in America. We conclude that they are essentially different. 相似文献
11.
Tao Lin Qing Yang Xu Zhang 《Numerical Methods for Partial Differential Equations》2015,31(6):1925-1947
We present partially penalized immersed finite element methods for solving parabolic interface problems on Cartesian meshes. Typical semidiscrete and fully discrete schemes are discussed. Error estimates in an energy norm are derived. Numerical examples are provided to support theoretical analysis. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1925–1947, 2015 相似文献
12.
We study the superconvergence property of fully discrete finite element approximation for quadratic optimal control problems governed by semilinear parabolic equations with control constraints. The time discretization is based on difference methods, whereas the space discretization is done using finite element methods. The state and the adjoint state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions. First, we define a fully discrete finite element approximation scheme for the semilinear parabolic control problem. Second, we derive the superconvergence properties for the control, the state and the adjoint state. Finally, we do some numerical experiments for illustrating our theoretical results. 相似文献
13.
Jichun Li 《Numerical Methods for Partial Differential Equations》2006,22(4):884-896
By using a special interpolation operator developed by Girault and Raviart (finite element methods for Navier‐Stokes Equations, Springer‐Verlag, Berlin, 1986), we prove that optimal error bounds can be obtained for a fourth‐order elliptic problem and a fourth‐order parabolic problem solved by mixed finite element methods on quasi‐uniform rectangular meshes. Optimal convergence is proved for all continuous tensor product elements of order k ≥ 1. A numerical example is provided for solving the fourth‐order elliptic problem using the bilinear element. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006 相似文献
14.
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
15.
《Numerical Methods for Partial Differential Equations》2018,34(3):838-856
In this paper, the author derives an ‐superconvergence for the piecewise linear Ritz‐Galerkin finite element approximations for the second‐order elliptic equation equipped with Dirichlet boundary conditions. This superconvergence error estimate is established between the finite element solution and the usual Lagrange nodal point interpolation of the exact solution, and thus the superconvergence at the nodal points of each element. The result is based on a condition for the finite element partition characterized by the coefficient tensor and the usual shape functions on each element, called ‐equilateral assumption in this paper. Several examples are presented for the coefficient tensor and finite element triangulations which satisfy the conditions necessary for superconvergence. Some numerical experiments are conducted to confirm this new theory of superconvergence. 相似文献
16.
Shimin Chai Yongkui Zou Chenguang Zhou Wenju Zhao 《Numerical Methods for Partial Differential Equations》2019,35(5):1745-1755
This paper is devoted to a newly developed weak Galerkin finite element method with the stabilization term for a linear fourth order parabolic equation, where weakly defined Laplacian operator over discontinuous functions is introduced. Priori estimates are developed and analyzed in L2 and an H2 type norm for both semi‐discrete and fully discrete schemes. And finally, numerical examples are provided to confirm the theoretical results. 相似文献
17.
In this paper, we consider the a posteriori error analysis of discontinuous Galerkin finite element methods for the steady and nonsteady first order hyperbolic problems with inflow boundary conditions. We establish several residual-based a posteriori error estimators which provide global upper bounds and a local lower bound on the error. Further, for nonsteady problem, we construct a fully discrete discontinuous finite element scheme and derive the a posteriori error estimators which yield global upper bound on the error in time and space. Our a posteriori error analysis is based on the mesh-dependent a priori estimates for the first order hyperbolic problems. These a posteriori error analysis results can be applied to develop the adaptive discontinuous finite element methods. 相似文献
18.
Higher‐order finite volume element methods based on Barlow points for one‐dimensional elliptic and parabolic problems
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Min Yang 《Numerical Methods for Partial Differential Equations》2015,31(4):977-994
The article is devoted to a kind of higher‐order finite volume element methods, where the dual partitions are constructed by Barlow points, for elliptic and parabolic problems in one space dimension. Techniques to derive the stability and to control the nonsymmetry are presented. Superconvergence and the optimal order errors in the H1‐ and L2‐norms are obtained. Numerical results illustrate the theoretical findings. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 977–994, 2015 相似文献
19.
We investigate a new hybrid method we call the discontinuous Galerkin‐inverted finite element method (DGIFEM) to approximate the solution of elliptic problems in , especially when the growth or the decay of the solution is very slow. On the basis of both the discontinuous Galerkin discretization and the inverted finite element method, the DGIFEM keeps part of the domain bounded and maps the other infinite extent into a bounded region via a suitable polygonal inversion. The numerical solution is then constructed in an appropriate subspace of weighted Sobolev spaces, where the weights allow the control of the growth or the decay of functions at infinity. A careful study of the convergence of the DGIFEM is carried out and shows that the optimal order of convergence can always be reached. Finally, some numerical results are given as illustration of the good performance of the proposed method. 相似文献
20.
Analysis of least-squares mixed finite element methods for nonlinear nonstationary convection-diffusion problems 总被引:6,自引:0,他引:6
Dan-Ping Yang. 《Mathematics of Computation》2000,69(231):929-963
Some least-squares mixed finite element methods for convection-diffusion problems, steady or nonstationary, are formulated, and convergence of these schemes is analyzed. The main results are that a new optimal a priori error estimate of a least-squares mixed finite element method for a steady convection-diffusion problem is developed and that four fully-discrete least-squares mixed finite element schemes for an initial-boundary value problem of a nonlinear nonstationary convection-diffusion equation are formulated. Also, some systematic theories on convergence of these schemes are established.