Superconvergence.bib

A.B. Andreev and R.D. Lazarov. Error estimate of type superconvergence of the gradient for quadratic triangular elements. C.R. Acad. Bulgare Sci., 36:1179-1182, 1984.

A.B. Andreev and R.D. Lazarov. Superconvergence of the gradient for quadratic triangular finite elements. Numer. Methods Partial Differential equations, 4:15-32, 1988.

Ivo Babuška and W. C. Rheinboldt. A posteriori error error estimates for the finite element method. International Journal for Numerical Methods in Engineering, 12:1597--1615, 1978.

I.Babuska and T.Strouboulis and C.S.Upadhyay. $\eta$-superconvergence of finite element approximations in the interior of general meshes of triangles. Comput. Methods Appl. Mech. Engrg., 122:273--305, 1995.

R. E. Bank and Jinchao Xu. Asymptotically Exact A Posteriori Error Estimators, {P}art {II}: General Unstructured Grids. SIAM Journal on Numerical Analysis, 41:2313-2332, 2003.

Bank.R;Xu.J2003

ABSTRACT: In Part I of this work [SIAM Journal on Numer. Anal. , 41 (2003), pp. 2294--2312], we analyzed superconvergence for piecewise linear finite element approximations on triangular meshes where most pairs of triangles sharing a common edge form approximate parallelograms. In this work, we consider superconvergence for general unstructured but shape regular meshes. We develop a postprocessing gradient recovery scheme for the finite element solution uh, inspired in part by the smoothing iteration of the multigrid method. This recovered gradient superconverges to the gradient of the true solution and becomes the basis of a global a posteriori error estimate that is often asymptotically exact. Next, we use the superconvergent gradient to approximate the Hessian matrix of the true solution and form local error indicators for adaptive meshing algorithms. We provide several numerical examples illustrating the effectiveness of our procedures.

R. E. Bank and Jinchao Xu. Asymptotically Exact A Posteriori Error Estimators, {P}art {I}: Grids with Superconvergence. SIAM Journal on Numerical Analysis, 41:2294-2312, 2003.

Bank.R;Xu.J2003a

ABSTRACT: In Part I of this work, we develop superconvergence estimates for piecewise linear finite element approximations on quasi-uniform triangular meshes where most pairs of triangles sharing a common edge form approximate parallelograms. In particular, we first show a superconvergence of the gradient of the finite element solution uh and to the gradient of the interpolant $u_I$. We then analyze a postprocessing gradient recovery scheme, showing that $Q_h\nabla u_h$ is a superconvergent approximation to $\nabla u$. Here Qh is the global L2 projection. In Part II, we analyze a superconvergent gradient recovery scheme for general unstructured, shape regular triangulations. This is the foundation for an a posteriori error estimate and local error indicators.

J. Brandts and M. K\v{r}\'{i}\v{z}ek. History and Future of Superconvergnece in Three-Dimensional Finite Element Methods. Proc. Conf. Finite Element Methods: Three dimensional Problems, GAKUTO Internat. Series Math. Sci. Appl., 15:22--33, 2001.

Brandts.J;Krizek.M2001

ABSTRACT: We will give an overview of superconvergnece results for finite element methods applied to problems in three space dimensions. Apart from that, we sketch techniques that could be applied to three dimensional superconvergnece questions, and indicate what exactly makes the three-dimensional case so much harder to tackle than the two-dimensional case, for which many more results are known.

J. Brandts and M. K\v{r}\'{i}\v{z}ek. Gradient superconvergence on uniform simplicial partitions of polytopes. IMA Journal of Numerical Analysis, 23:489-505, 2003.

J. Brandts and M. K\v{r}\'{i}\v{z}ek. Superconvergence Of Tetrahedral Quadratic Finite Elements. Journal of Computational Mathematics, 23:27-36, 2005.

Jan H. Brandts. Superconvergence and a posteriori error estimation for triangular mixed finite elements. Numerische Mathematik, 68:311-324, 1994.

Jan H. Brandts. Superconvergence phenomena in finite element methods. , :, 1994.

Brandts, Jan H.. Superconvergence for triangular order k=1 Raviart-Thomas mixed finite elements and for triangular standard quadratic finite element methods. Applied Numerical Mathematics, 34:39--58, 2000.

Brandts.J2000

ABSTRACT: We will prove, for a model problem, that on regular families of uniform triangulations, the vector variable of the order k=1 Raviart-Thomas type mixed finite element method, is superconvergent with respect to Fortin interpolation. For lowest order k=0 this was already proved in (Brandts, 1994). As a side product of the present analysis, we obtain similar results for the gradient of the standard quadratic finite element method, also with respect to Fortin interpolation.Although the use of Fortin interpolation instead of Lagrange interpolation in the setting of standard finite elements is somewhat unusual, it turns out that the superconvergence for standard quadratic elements with respect to Lagrange interpolation, proved in (Goodsell and Whiteman, 1991), is a direct corollary of it. As a result, the post-processing scheme that was developed in (Goodsell and Whiteman, 1991) to raise the approximation order of the gradient of the standard finite element approximation, can be adapted to improve the approximation quality of the mixed finite element vector variable in a similar fashion.The Fortin interpolation approach results moreover in L2({$[$}Omega{$]$})-superconvergence for the scalar variable.

Hongsen Chen and Junping Wang. An Interior Estimate of Superconvergence for Finite Element Solutions for Second-Order Elliptic Problems on Quasi-uniform Meshes by Local Projections. SIAM Journal on Numerical Analysis, 41:1318--1338, 2003.

Chen.H;Wang.J2003

ABSTRACT: This paper establishes some superconvergence estimates for finite element solutions of second-order elliptic problems by a projection method depending only on local properties of the domain and the finite element solution. The projection method is a postprocessing procedure that constructs a new approximation by using the method of least squares. In particular, some local superconvergence estimates in the L2 and $L^\infty$ norms are derived for the local projections of the Galerkin finite element solution. The results have two prominent features. First, they are established for any quasi-uniform meshes, which are of practical importance in scientific computation. Second, they are derived on the basis of local properties of the domain and the solution for the second-order elliptic problem. Therefore, the result of this paper can be employed to provide useful a posteriori error estimators in practical computing.

Long Chen. Superconvergence of tetrahedral linear finite elements. International Journal of Numerical Analysis and Modeling, 3:273--282, 2006.

Chen.L2006

ABSTRACT: In this paper, we show that the piecewise linear finite element solution $u_{h}$ and the linear interpolation $u_{I}$ have superclose gradient for tetrahedral meshes, where most elements are obtained by dividing approximate parallelepiped into six tetrahedra. We then analyze a post-processing gradient recovery scheme, showing that the global $L^2$ projection of $\nabla u_h$ is a superconvergent gradient approximation to $\nabla u$.

C. Dawson and C. Woodward and M. Wheeler. A two-grid finite difference scheme for nonlinear parabolic equations. SIAM Journal on Numerical Analysis, 35:435-452, 1998.

Dawson.C;Woodward.C;Wheeler.M1998

ABSTRACT: We present a two-level finite difference scheme for the approximation of nonlinear parabolic equations. Discrete inner products and the lowest-order Raviart--Thomas approximating space are used in the expanded mixed method in order to develop the finite difference scheme. Analysis of the scheme is given assuming an implicit time discretization. In this two-level scheme, the full nonlinear problem is solved on a "coarse" grid of size H. The nonlinearities are expanded about the coarse grid solution and an appropriate interpolation operator is used to provide values of the coarse grid solution on the fine grid in terms of superconvergent node points. The resulting linear but nonsymmetric system is solved on a "fine" grid of size h. Some a priori error estimates are derived which show that the discrete L\infty(L2) and L2(H1) errors are $O(h^2 + H^{4-d/2} + \Delta t)$, where $d \geq 1$ is the spatial dimension.

Richard E. Ewing and Michael M. Liu and Junping Wang. Superconvergence of Mixed Finite Element Approximations over Quadrilaterals. SIAM Journal on Numerical Analysis, 36:772-787, 1999.

Richard E. Ewing and Mingjun Liu and Junping Wang. A New Superconvergence for Mixed Finite Element Approximations. SIAM Journal on Numerical Analysis, 40:2133-2150, 2002.

Q. Fang and T. Yamamoto. Superconvergence of finite difference approximations for convection-diffusion problems. Numer. Lin. Alg. Appl., 8:99--110, 2001.

Fierro, Francesca and Veeser, Andreas. A posteriori error estimators, gradient recovery by averaging, and superconvergence. Numerische Mathematik, 103:267--298, 2006.

Fierro.F;Veeser.A2006

ABSTRACT: For the linear finite element solution to a linear elliptic model problem, we derive an error estimator based upon appropriate gradient recovery by local averaging. In contrast to popular variants like the ZZ estimator, our estimator contains some additional terms that ensure reliability also on coarse meshes. Moreover, the enhanced estimator is proved to be (locally) efficient and asymptotically exact whenever the recovered gradient is superconvergent. We formulate an adaptive algorithm that is directed by this estimator and illustrate its aforementioned properties, as well as their importance, in numerical tests.

G. Goodsell and J. R Whiteman. Superconvergence of recovered gradients of piecewise quadratic finite element approximations. Numer. Methods Partial Differential equations, 7:85-99, 1991.

G. Goodsell. Pointwise superconvergence of the gradient for the linear tetrahedral element. Numer. Methods Partial Differential Equations, 10:651-666, 1994.

B. Heimsund and X. Tai and J. Wang. Superconvergence for the gradient of finite element approximations by ${L}^2$ projections. SIAM Journal on Numerical Analysis, 40:1263--1280, 2002.

Ivan. Hlavacek and Michal K\v{r}\'{i}\v{z}ek. On a superconvergent finite element scheme for elliptic systems, I Dirichlet boundary condition. Aplikace Matematiky, 32:131--154, 1987.

W. Hoffmann and A. H. Schatz and L. B. Wahlbin and G. Wittum. Asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes {I}: A smooth problem and globally quasi-uniform meshes. Mathematics of Computation, 70:897-909, 2001.

Hoffmann.W;Schatz.A;Wahlbin.L2001

ABSTRACT: A class of a posteriori estimators is studied for the error in the maximum-norm of the gradient on single elements when the finite element method is used to approximate solutions of second order elliptic problems. The meshes are unstructured and, in particular, it is not assumed that there are any known superconvergent points. The estimators are based on averaging operators which are approximate gradients, $recovered$ $gradients$, which are then compared to the actual gradient of the approximation on each element. Conditions are given under which they are asympotically exact or equivalent estimators on each single element of the underlying meshes. Asymptotic exactness is accomplished by letting the approximate gradient operator average over domains that are large, in a controlled fashion to be detailed below, compared to the size of the elements.

Y. Q. Huang and J. Xu. Superconvergence for quadratic triangular finite elements on mildly structured grids. Preprint, :, 2005.

V. K. Kantchev and R. D. Lazarov. Superconvergence of the gradient of linear finite elements for 3-D {P}ossion equation. Proceedings of the Conference on Optimal Algorithms, :172-182, 1986.

M. K\v{r}\'{i}\v{z}ek and P. Neittanmki. Superconvergence phenomenon in the finite element method arising from averaging gradients. Numerische Mathematik, 45:105--116, 1984.

M. K\v{r}\'{i}\v{z}ek. Superconvergence Phenomena on Three-Dimensional Meshes. International Journal of Numerical Analysis and Modeling, 2:43-56, 2005.

A. M. Lakhany and I. Marek and J. R. Whiteman. Superconvergence results on mildly structured triangulations. Comput. Methods Appl. Mech. Engrg., 189:1-75, 2000.

B. Li and Z. M. Zhang. Analysis of a class of superconvergence patch recovery techniques for linear and bilinear finite elements. Numerical Methods for Partial Differential Equations, 15:151-167, 1999.

Li.B;Zhang.Z1999

ABSTRACT: Mathematical proofs are presented for the derivative superconvergence obtained by a class of patch recovery techniques for both linear and bilinear finite elements in the approximation of elliptic second-order problems. In detail, the authors consider both the locally discrete least-squares recovery and the traditional post-processing technique by local $L$ projection recovery for both triangular linear and rectangular bilinear finite element approximations of general second-order elliptic problems on two-dimensional convex polygonal domains. They prove that the derivative superconvergence is achieved by both methods for the triangular linear element on a strongly regular family of meshes, by the locally discrete least-squares recovery for the rectangular bilinear element on a quasi-uniform family of meshes, and by the local $L$ projection recovery for the rectangular bilinear element on a unidirectionally uniform family of meshes. A one-dimensional example is also given.

B. Li. Superconvergence for higher-order triangular finite elements. Chinese J Numer Math Appl, 12:75--79, 1990.

Bo Li. Lagrange interpolation and finite element superconvergence. Numerical Methods for Partial Differential Equations, 20:33--59, 2004.

Jichun Li and MARY F. Wheeler. Uniform Convergence and Superconvergence of Mixed Finite Element Methods on Anisotropically Refined Grids. SIAM Journal on Numerical Analysis, 38:770--798, 2000.

Li.J;Wheeler.M2000

ABSTRACT: The lowest order Raviart--Thomas rectangular element is considered for solving the singular perturbation problem $-\mbox{div}(a\nabla p)+bp=f,$ where the diagonal tensor $a=(\varepsilon^2,1)$ or $a=(\varepsilon^2,\varepsilon^2).$ Global uniform convergence rates of O(N -1) for both p and a 1/2\nabla p$ in the L 2-norm are obtained in both cases, where N is the number of intervals in either direction. The pointwise interior (away from the boundary layers) convergence rates of O(N -1) for p are also proved. Superconvergence (i.e., O(N -2)) at special points and O(N -2) global L 2 estimate for both p and $a^{1/2}\nabla p$ are obtained by a local postprocessing. Numerical results support our theoretical analysis. Moreover, numerical experiments show that an anisotropic mesh gives more accurate results than the standard global uniform mesh.

Li, Jichun. Convergence and superconvergence analysis of finite element methods on highly nonuniform anisotropic meshes for singularly perturbed reaction-diffusion problems. Applied Numerical Mathematics, 36:129--154, 2001.

Q. Lin and Jinchao Xu. Linear finite elements with high accuracy. J. Comp. Math., 3:115--133, 1985.

Ahmed Naga and Zhimin Zhang. A Posteriori Error Estimates Based on the Polynomial Preserving Recovery. SIAM Journal on Numerical Analysis, 42:1780--1800, 2004.

Naga.A;Zhang.Z2004

ABSTRACT: Superconvergence of order $O(h^{1+\rho})$, for some $\rho > 0$, is established for the gradient recovered with the polynomial preserving recovery (PPR) when the mesh is mildly structured. Consequently, the PPR-recovered gradient can be used in building an asymptotically exact a posteriori error estimator.

Jeffrey S. Ovall. Asymptotically exact functional error estimators based on superconvergent gradient recovery. Numerische Mathematik, 102:543--558, 2006.

Ovall.J2006

ABSTRACT: The use of dual/adjoint problems for approximating functionals of solutions of PDEs with great accuracy or to merely drive a goal-oriented adaptive refinement scheme has become well-accepted, and it continues to be an active area of research. The traditional approach involves dual residual weighting (DRW). In this work we present two new functional error estimators and give conditions under which we can expect them to be asymptotically exact. The first is of DRW type and is derived for meshes in which most triangles satisfy an -approximate parallelogram property. The second functional estimator involves dual error estimate weighting (DEW) using any superconvergent gradient recovery technique for the primal and dual solutions. Several experiments are done which demonstrate the asymptotic exactness of a DEW estimator which uses a gradient recovery scheme proposed by Bank and Xu, and the effectiveness of refinement done with respect to the corresponding local error indicators.

G. H. Paulino and I. F. M. Menezes and J. B. Cavalcante Neto and L. F. Martha. A methodology for adaptive finite element analysis: Towards an integrated computational environment. Comutational Mechanics, 23:361-388, 1999.

Paulino.G;Menezes.I;Neto.J1999

ABSTRACT: This work introduces a methodology for self-adaptive numerical procedures, which relies on the various components of an integrated, object-oriented, computational environment involving pre-, analysis, and post-processing modules. A basic platform for numerical experiments and further development is provided, which allows implementation of new elements/error estimators and sensitivity analysis. A general implementation of the Superconvergent Patch Recovery (SPR) and the recently proposed Recovery by Equilibrium in Patches (REP) is presented. Both SPR and REP are compared and used for error estimation and for guiding the adaptive remeshing process. Moreover, the SPR is extended for calculating sensitivity quantities of first and higher orders. The mesh (re-)generation process is accomplished by means of modern methods combining quadtree and Delaunay triangulation techniques. Surface mesh generation in arbitrary domains is performed automatically (i.e. with no user intervention) during the self-adaptive analysis using either quadrilateral or triangular elements. These ideas are implemented in the Finite Element System Technology in Adaptivity (FESTA) software. The effectiveness and versatility of FESTA are demonstrated by representative numerical examples illustrating the interconnections among finite element analysis, recovery procedures, error estimation/adaptivity and automatic mesh generation.

N. Robidoux. Polynomial Histopolation, Superconvergent Degrees of Freedom, and Pseudospectral Discrete Hodge Operators. Preprint, :, 2000.

H. Roos and T. Lin{ß}. Gradient Recovery for Singularly Perturbed Boundary Value Problems {I}: One-Dimensional Convection-Diffusion. Computing, 66:163-178, 2001.

Roos.H;Linss.T2001

ABSTRACT: We consider a Galerkin finite element method that uses piecewise linears on a class of Shishkin-type meshes for a model singularly perturbed convection-diffusion problem. We pursue two approaches in constructing superconvergent approximations of the gradient. The first approach uses superconvergence points for the derivative, while the second one combines the consistency of a recovery operator with the superconvergence property of an interpolant. Numerical experiments support our theoretical results.

A. Schatz and I. Sloan and L. Wahlbin. Superconvergence in Finite Element Methods and Meshes That are Locally Symmetric with Respect to a Point. SIAM Journal on Numerical Analysis, 33:505--521, 1996.

Schatz.A;Sloan.I;Wahlbin.L1996

ABSTRACT: Consider a second-order elliptic boundary value problem in any number of space dimensions with locally smooth coefficients and solution. Consider also its numerical approximation by standard conforming finite element methods with, for example, fixed degree piecewise polynomials on a quasi-uniform mesh-family (the ``$h$-method''). It will be shown that, if the finite element function spaces are locally symmetric about a point $x_0 $ with respect to the antipodal map $x \to x_0 - (x - x_0 )$, then superconvergence ensues at xo under mild conditions on what happens outside a neighborhood of $x_0 $. For piecewise polynomials of even degree, superconvergence occurs in function values; for piecewise polynomials of odd degree, it occurs in derivatives.

Alfred H. Schatz. Perturbations of Forms and Error Estimates for the Finite Element Method at a Point, with an Application to Improved Superconvergence Error Estimates for Subspaces that Are Symmetric with Respect to a Point. SIAM Journal on Numerical Analysis, 42:2342--2365, 2005.

Schatz.A2005

ABSTRACT: We first derive a variety of local error estimates for u - uh at a point x0, where uh belongs to a finite element space Shr and is an approximation to u satisfying the local equations $A(u-u_h,\varphi) = F(\varphi)$ for all $\varphi$ in Shr with compact support in a neighborhood of x0. Here the $A(\cdot,\cdot)$ are bilinear forms associated with second order elliptic equations and the F are linear functionals. In the case that $F \equiv 0$ our results coincide with those of Schatz [SIAM Journal on Numer. Anal., 38 (2000), pp. 1269--1293] but are improvements when $F \neq 0$. We apply these results to improve the superconvergence error estimates obtained by Schatz, Sloan, and Wahlbin [SIAM Journal on Numer. Anal., 33 (1996), pp. 505--521] at points x0 where the subspaces are symmetric with respect to x0.

Shi Shu and Cunyun Nie and Haiyun Yu and Yunqing Huang. A preserving-symmetry finite volume scheme and superconvergence on quadrangle grids. Priprint, :, 2005.

V. Thomee and N. Zhang and Jinchao Xu. Superconvergence of gradient in piecewise linear finite element approximation to a parabolic problem. SIAM Journal on Numerical Analysis, 26:553--573, 1989.

Lars B. Wahlbin. On Superconvergence up to boundaries in finite element methods: a counter example. SIAM Journal on Numerical Analysis, 29:937--946, 1992.

Wahlbin, Lars B.. General principles of superconvergence in {G}alerkin finite element methods. Finite element methods (Jyv{\''a}skyl{\''a}, 1997), :269--285, 1998.

Junping Wang and Xiu Ye. Superconvergence of Finite Element Approximations for the Stokes Problem by Projection Methods. SIAM Journal on Numerical Analysis, 39:1001--1013, 2001.

Wang.J;Ye.X2001

ABSTRACT: This paper derives a general superconvergence result for finite element approximations of the Stokes problem by using projection methods proposed and analyzed recently by Wang [J. Math. Study, 33 (2000), pp. 229--243] for the standard Galerkin method. The superconvergence result is based on some regularity assumption for the Stokes problem and is applicable to any finite element method with regular but nonuniform partitions. The method is proved to give a convergent scheme for certain finite element spaces which fail to satisfy the well-known uniform inf-sup condition of Brezzi and Babuska.

Junping Wang. A superconvergence analysis for finite element solutions by the least-squares surface fitting on irregular meshes for smooth problems. J. Math. Study, 33:229--243, 2000.

B. I. Wohlmuth and R. H. W. Hoppe. A comparison of a posteriori error estimators for mixed finite element discretizations by Raviart-Thomas elements. Mathematics of Computation, 82:253-279, 1999.

Wohlmuth.B;Hoppe.R1999

ABSTRACT: We consider mixed finite element discretizations of linear second order elliptic boundary value problems with respect to an adaptively generated hierarchy of possibly highly nonuniform simplicial triangulations. In particular, we present and analyze four different kinds of error estimators: a residual based estimator, a hierarchical one, error estimators relying on the solution of local subproblems and on a superconvergence result, respectively. Finally, we examine the relationship between the presented error estimators and compare their local components.

Jiming Wu and Weiwei Sun. The Superconvergence of the Composite Trapezoidal Rule for Hadamard Finite Part Integrals. Numerische Mathematik, 102:343 - 363, 2005.

Wu.J;Sun.W2005

ABSTRACT: The composite trapezoidal rule has been well studied and widely applied for numerical integrations and numerical solution of integral equations with smooth or weakly singular kernels. However, this quadrature rule has been less employed for Hadamard finite part integrals due to the fact that its global convergence rate for Hadamard finite part integrals with (p+1)-order singularity is p-order lower than that for the Riemann integrals in general. In this paper, we study the superconvergence of the composite trapezoidal rule for Hadamard finite part integrals with the second-order and the third-order singularity, respectively. We obtain superconvergence estimates at some special points and prove the uniqueness of the superconvergence points. Numerical experiments confirm our theoretical analysis and show that the composite trapezoidal rule is efficient for Hadamard finite part integrals by noting the superconvergence phenomenon.

Jinchao Xu and Z. M. Zhang. Analysis of recovery type a posteriori error estimators for mildly structured grids. Mathematics of Computation, :781--801, 2003.

Xu.J;Zhang.Z2003

ABSTRACT: Some recovery type error estimators for linear finite elements are analyzed under $O(h^{1+\alpha}) (\alpha > 0)$ regular grids. Superconvergence of order $O(h^{1+\rho}) (0 < \rho\leq \alpha)$ is established for recovered gradients by three different methods. As a consequence, a posteriori error estimators based on those recovery methods are asymptotically exact.

Jinchao Xu. The error analysis and the improved algorithms for the infinite element method. Proceedings of the 1984 Beijing Symposium on Differential Geometry and Differential Equations, :326--331, 1985.

Xu.J1985

ABSTRACT: This paper is concerned with the theoretical analysis and the practiccal algorithms for the infinite element method which was developed in recent years. With the aid of the inequalities related to the Sobolev space given in the paper, the error estimates in the norms of ____ (with weight), and ___ and ___, the estimates like superconvergence and so on are comprehensively investigated for the method. Many of the conclusions obtained are optimal or nearly optimal. Also, some relative algorithms are analyzed and improved and the results are suported by numerical experiments

Zhimin Zhang and Ahmed Naga. Validation of the a posteriori error estimator based on polynomial preserving recovery for linear elements. Internat. J. Numeri. Methods Engrg., 61:1860--1893, 2004.

Zhang.Z;Naga.A2004

ABSTRACT: The purpose of this work is to investigate the quality of the a posteriori error estimator based on polynomial preserving recovery (PPR). The main tool in this investigation is the computer-based theory. Also, a comparison is made between this estimator and the one based on superconvergence patch recovery (SPR). The results of this comparison are found to be in favour of the estimator based on PPR.

Z. M. Zhang and A. Naga. A new finite element gradient recovery method: Superconvergence property. SIAM Journal on Scientific Computing, 26:1192--1213, 2005.

Zhang.Z;Naga.A2005

ABSTRACT: This is the first in a series of papers in which a new gradient recovery method is introduced and analyzed. It is proved that the method is superconvergent for translation invariant finite element spaces of any order. The method maintains the simplicity, efficiency, and superconvergence properties of the Zienkiewicz--Zhu patch recovery method. In addition, for uniform triangular meshes, the method is superconvergent for the linear element under the chevron pattern, and ultraconvergent at element edge centers for the quadratic element under the regular pattern. Applications of this new gradient recovery technique will be discussed in forthcoming papers

Zhang, Zhimin and Zhu, J. Z.. Superconvergence of the derivative patch recovery technique and a posteriori error estimation. Modeling, mesh generation, and adaptive numerical methods for partial differential equations (Minneapolis, MN, 1993), :431--450, 1995.

Z. M. Zhang. Derivative superconvergent points in finite element solutions of Poisson's equation for the serendipity and intermediate families - a theoretical justification. Mathematics of Computation, 67:541-552, 1998.

Zhang.Z1998

ABSTRACT: Finite element derivative superconvergent points for the Poisson equation under local rectangular mesh (in the two dimensional case) and local brick mesh (in the three dimensional sitNumer. Meth. PDEsuation) are investigated. All superconvergent points for the finite element space of any order that is contained in the tensor-product space and contains the intermediate family can be predicted. In case of the serendipity family, the results are given for finite element spaces of order below 7. Any finite element space that contains the complete polynomial space will have at least all superconvergent points of the related serendipity family.

Z. M. Zhang. Finite Element Superconvergence Approximation for One-Dimensional Singularly Perturbed Problems. Numer. Meth. PDEs, 18:374-395, 2002.

Zhang.Z2002

ABSTRACT: Superconvergence approximations of singularly perturbed two-point boundary value problems of reactiondiffusion 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.

Z. M. Zhang. Finite element superconvergence on {Shishkin} mesh for 2-{D} convection-diffusion problems. Mathematics of Computation, 72:1147-1177, 2003.

Zhang.Z2003

ABSTRACT: 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 O(N?2 ln 2 N + N?1. 5 ln N)inadiscrete -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 O(N?2 ln 2 N) is sharp for the boundary layer terms. As a by-product, an -uniform convergence of the same order is obtained for the L2-norm. Furthermore, under the same regularity assumption, an -uniform convergence of order N?3/2 ln 5/2 N + N?1 ln1/2 N in the norm is proved for some mesh points in the boundary layer region.

G. Zhou and R. Rannacher. Pointwise superconvergence of the streamline diffusion finite element method. Numer. Meth. PDEs, 12, CMP 96:05:123-145, 1996.

Zhu, J. Z. and Zienkiewicz, O. C.. Superconvergence recovery technique and a posteriori error estimators. International Journal for Numerical Methods in Engineering, 30:1321--1339, 1990.

Qiding Zhu AND Qun Lin. Finite element superconvergence theory. Hunan Science Press, :, 1989.

Qiding Zhu. Derivatives superconvergence points for triangular quadratic finite elements. , 120:36-45, 1993.

O. C. Zienkiewicz and J. Z. Zhu. The superconvergence patch recovery and a posteriori error estimates. {P}art 2: Error estimates and adaptivity. International Journal for Numerical Methods in Engineering, 33:1365-1382, 1992.

O. C. Zienkiewicz and J. Z. Zhu. The superconvergence patch recovery and a posteriori error estimates. {P}art 1: The recovery techniques. International Journal for Numerical Methods in Engineering, 33:1331-1364, 1992.

M. Zlamal. Some superconvergence results in the finite element method. Mathematical Aspects of Finite Element Methods, :, 1975.

M. Zlamal. Superconvergence and reduced integration in the finite element method. Mathematics of Computation, 32:663--685, 1978.