Title: Robust A Posteriori Error Estimation for Non-Conforming Finite Element Approximation

Abstract: We consider the problem of developing computable upper bound error estimators for non-conforming finite element approximation of linear second order elliptic equations where the permeability coefficient is allowed to undergo large jumps in value across interfaces between differing media.  An error estimator is derived and shown to provide a computable upper bound on the error, and, up to a constant depending only on the geometry, also provides a local lower bound on the error.  Moreover, we consider the behavior of the estimator when the magnitude of the jumps in permeability is varied and establish quantitative bounds for the dependence of the constants on the jumps.  The extension of these ideas to non-conforming finite element approximation of the Stokes in equations is also described.  Illustrative numerical examples are included.

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Title: Superconvergence in Least-Squares Mixed Finite Element Methods

Abstract: In this presentation, we establish an analysis of the Least Squares Mixed Finite Element Methods in terms of Standard and Mixed elements respectively.  From this analysis we derive conditions for the LSMFE to exhibit superconvergence.  As a side product of the analysis, we derive a strengthened Cauchy-Schwarz inequality that enables easy derivation of the coercivity of the LSMFE bilinear form.  Even though this coercivity result is not new, its proof is much simpler and more transparent than the existing proofs.  This is joint work with Yanping Chen, Xiangtan University.

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Title: Averaging (FEM) 4 Everything?

Abstract: The striking simplicity of averaging techniques and their amazing accuracy in too many numerical examples made them an extremely popular tool in scientific computing whenever finite elements might be useful.  Given a discrete flux p_h and an easily post-processed approximation A p_h to compute the error estimator h_A : = || p_h -  A p_h||.  One does not even need an equation to employ that technique occasionally named after Zienkiewicz & Zhu.

The beginning of a mathematical justification of the error estimator h_A  as a computable approximation of the (unknown) error || p - p_h|| involved the concept of super-convergence points.  For highly structured meshes and a very smooth exact solution p, the error || p -  A p_h|| of the post-processed approximation A p_h may be (much) smaller than || p - p_h|| of the given p_h.  Under the assumption that || p -  A p_h|| =  h.o.t. is relatively sufficiently small, the triangle inequality immediately verifies reliability, i.e.,

 || p - p_h|| £ C_rel  h_A + h.o.t.,

and efficiency, i.e.,

h_A  £ C_eff || p - p_h|| + h.o.t.,

of the averaging error estimator h_A.  However, the underlying assumptions essentially contradict the notion of adaptive grid refining for optimal experimental convergence rates when p is singular. Moreover, the proper treatment of boundary
conditions lacks a serious inside.

The presentation reports on old and new arguments for reliability and efficiency in the above sense with multiplicative
constants C_rel and C_eff and higher order terms h.o.t.  Hi-lighted are the general class of meshes, averaging techniques, or finite element methods (conforming, nonconforming, and mixed elements) for elliptic PDEs. Numerical examples illustrate the amazing accuracy of h_A.  The presentation closes with a discussion on current developments and the limitations as well as the perspectives of averaging techniques.

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Title: Orthogonal Correction Technique in Superconvergence Analysis

Abstract: 

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Title: Some Researches on Mixed Finite Element Method

Abstract: This report summarizes our recent researches on mixed finite element method.
1. For pursing a high performance of mixed finite element algorithm to simulate the convection-dominated problems, we try to combine mixed finite element with the characteristics method.  The idea is used to simulate the linear, nonlinear convection-dominated problems; convection-dominated integrodi differential equations of parabolic type.  Numerical analysis and numerical tests are presented. 
2. The regularized long wave equation and Sobolev equation are simulated by H¹- Galerkin mixed finite element method.  Compared to the standard H¹ Galerkin, C¹-continuity for the approximating finite dimensional subspaces can be relaxed for the proposed method.  Moreover, it is shown that the finite element approximations have the same rates of convergence as in the classical mixed finite element method, but without LBB consistency. 
3. The expanded mixed finite element method is employed to solve the problems such as the linear, nonlinear parabolic equations; the linear, nonlinear hyperbolic equations.  This formulation expands the standard mixed formulation in the sense that three variables are explicitly treated; i.e., the scalar unknown, its gradient and its flux (the coefficient times the gradient).  It is suitable for the case where the coefficient of the differential equations is a small tensor and does not need to be inverted. 

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Title: The Role of Superconvergence in Adaptive Finite Element Method

Abstract: This talk has three parts to show the important role of superconvergence in adaptive finite element method.  In the first part, some recent results about the superconvergence and postprocessing using techniques from multigrid on general unstructured grids will be reported.  Then a concept of optimal Delaunay triangulations for a given convex function will be introduced and an exact gradient recovery scheme will be derived on an optimal Delaunay triangulation.  The convection dominated problem will be studied in the third part.  It turns out that the superconvergence in the smooth part is crucial to get full convergence rate.

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Title: A posteriori Error Estimates of Mixed Finite Element Methods for Optimal Control

Abstract: In this talk, we present an a posteriori error analysis for mixed finite element approximation of convex optimal control problems.  We derive sharp a posteriori error estimates both for the state and control approximations under some assumptions which hold in many applications.  Such estimates, which are apparently not available in the literature, can be used to construct reliable adaptive remeshing schemes with mixed finite elements for control problems.

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Title: A posteriori Bound for Powell-Sabin-Henidl Elements and Related Superconvergence Properties

Abstract: In this paper we develop a posterior bounds for the approximation solutions based on divergence free Powell-Sabin-Henidl Elements to some flow problems involving Williamson and degenerate Carreau fluids.  The nonlinear structure of these fluids is relatively simple but the underlying natural function space is not.  Under certain regularity assumptions and by exploiting the structure of the lowest order divergence free PSH elements, we show that a posteriori error bounds may be established.

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Title: A posteriori Error Estimates for Parameter Estimation Problem in Equilibrium Equation

Abstract:

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Title: Metric Tensors for Anisotropic Mesh Generation

Abstract: It has been amply demonstrated that significant improvements in accuracy and efficiency can be gained when a properly chosen anisotropic mesh is used in the numerical solution a large class of problems which exhibit anisotropic solution features.  In practice, an anisotropic mesh is commonly generated as a quasi-uniform one in the metric determined by a tensor specifying the shape and size of elements.  It is crucial to choose an appropriate metric tensor.  In this talk I will present a new development for a general and mathematically rigorous formula of the metric tensor for use in any spatial dimension. The formulation is based on error estimates for polynomial preserving interpolation on simiplicial elements.  Two dimensional numerical results will be presented to demonstrate the ability of the metric tensor to produce anisotropic meshes with correct mesh concentration and good overall quality.  The procedure developed for defining the metric tensor can also be applied to other types of error estimates.

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Title: A Finite Difference Approach for the Superconvergence of Finite Element Methods

Abstract: In this talk we shall present some recent results on the superconvergence of finite element methods via a finite difference approach.  Some error expansions are obtained on Criss-Cross and Union-Jack meshes which were known as non three directional parallel meshes.  There are different structures on different kind of nodes.  Superconvergence and other post-processing techniques to increasing accuracy can be derived from the asymptotic expansion of the finite element approximations.

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Title: Superconvergence Phenomena on Three-dimensional Meshes

Abstract: In the first part we give an overview of superconvergence phenomena for the finite element method in three-dimensional space.  In particular, superconvergence results obtained for the solution and its gradient of second order boundary value problems of elliptic type will be presented.  We also survey typical difficulties with superconvergence on three-dimensional meshes, which do not arise in solving two-dimensional problems. For instance, one can prove the O(h⁴)-superconvergence at nodal points when solving the Poisson equation by linear elements over triangulations consisting solely of equilateral triangles.  Such a result cannot be generalized to the three-dimensional space, since the regular tetrahedron is not a space-filler.

In the second part we deal with superconvergence of linear and quadratic elements on tetrahedral meshes which are frequently used in applications.  The key idea is based on the fact that the gradient of the Ritz-Galerkin solution is superclose to the gradient of the Lagrange interpolation when uniform meshes are employed.  A special emphasize will be laid on an important subclass of basis functions for quadratic tetrahedral elements, namely we will examine properties of piecewise quadratic bubble functions.  Some numerical experiments confirming the theoretical superconvergence results will be presented as well. This is a joint work with Jan Brandts.

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Title: A posteriori Error Estimates for Particle Trajectories and Velocity in Stationary Stokes Flow

Abstract: We study numerically computed particle trajectories in stationary Stokes flow.  The velocity field is also computed numerically by a finite element method.  We prove a posteriori error estimates based on duality.  This work is motivated by our investigation of convective mixing of microflows.  This is joint work with my student Erik Svensson.

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Title: Optimal Uniform Convergence Analysis for A Two-dimensional Parabolic Problem with Two Small Parameters

Abstract: In this talk, we first give a brief overview about finite element analysis for singularly perturbed problems.  Then we consider a two-dimensional parabolic equation with two small parameters.  These small parameters make the underlying problem contains multiple scales spanning the whole problem domain.  By using the maximum principle with carefully chosen barrier functions, we obtain the pointwise derivative estimates of arbitrary order, from which an anisotropic mesh is constructed.  This mesh uses very finer mesh inside the small scale regions (where the boundary layers are located) than elsewhere (large scale regions).  A fully discrete backward difference Galerkin scheme based on this mesh with arbitrary order conforming tensor-product finite elements is discussed.  Note that the standard finite element
analysis technique can not be used directly for such highly nonuniform anisotropic meshes because of the violation of the quasi-uniformity assumption.  Then we use the integral identity superconvergence technique to prove the optimal uniform convergence in L²-norm.

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Title: Natural Superconvergent Points of 3D Finite Elements

Abstract: We analytically identify natural superconvergent points of function values and gradients for several common types three-dimensional polynomial finite elements.  Both the Poisson equation and the Laplace equation are discussed.  In particular, for the hexahedral and pentahedral elements, we studied Lagrangian and serendipity elements of order up to 6 and 5, respectively;  For the tetrahedral elements, we considered elements of order up to 4 in two different mesh patterns.  This is a joint work with Prof. Zhimin Zhang.

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Title: Recovery Based A posteriroir Error Estimators in Optimal Control

Abstract: In this talk we examine a class of error indicators formed through recovery in function values, in adaptive finite element approximation of optimal control.  We first look at some motivations and then some details of the indicators.  We then compare the performances of the indicators with residual based ones through numerical tests.

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Title: Some local weighted pointwise error estimates,both interior and up to the boundary, with applications.

Abstract: We discuss a variety of local estimates for the error in the finite element method for second order elliptic problems in R^N.  If time permits we shall discuss applications to 
1) A posteriore error estimates for the maximum norm of the gradient on each triangle for Dirichlet's problem on a nonconvex polygonal domain,
2) Some new superconvergence estimates for subspaces that are symmetric with respect to a point,
3) Local asymptotic expansions at similarity points of the mesh, and 
4) A new approach to Richardson extrapolation.

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Title: Some Local Weighted Pointwise Error Estimates, both Interior and up to the Boundary, with Applications

Abstract: We discuss a variety of local estimates for the error in the finite element method for second order elliptic problems in R^N.  If time permits we shall discuss applications to 1) A posteriore error estimates for the maximum norm of the gradient on each triangle for Dirichlet's problem on a nonconvex polygonal domain.  2) Some new superconvergence
estimates for subspaces that are symmetric with respect to a point.  3) Local asymptotic expansions at similarity points of the mesh, and 4) A new approach to Richardson extrapolation.

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Title: Some Superconveregence Results for Nonconforming Rotated Q_1 Elements

Abstract: For the nonconforming rotated Q_1 elements over midly distorted quadrilateral mesh, we propose a superconvergence property at the center, the vertices and the midpoints of four edges.  Numerics are presented to confirm this observation.

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Title: Superconvergence Analysis of Some Anisotropic Finite Elements

Abstract: It is well known that the regular assumption and quasi-uniform assumption on the triangulation are basic conditions in the analysis of FEMs.  However, the above assumptions are definite restrictions in FEMs' application.  For example, the solution of some problems may have anisotropic behavior in parts of the domain.  This means that the solution varies significantly only in certain directions, such as the diffusion problems in domains with edges and singularly perturbed convection-diffusion-reaction problem where boundary or interior layers appear.  In such cases, it is an obvious idea to reflect anisotropy in the discretization by using anisotropic meshes, with a small mesh size in the direction of the rapid variation of the solution and a larger mesh size in the perpendicular direction. 

The main aim of this report is to present the superconvergence results of some anisotropic finite elements based on some novel approaches and techniques, which include that of the nonconforming quasi-Wilson element, five parameter rectangular element, a new Hermite type rectangular element, a new Hermite type triangular element and the Bicubic Hermite rectangular element.  Numerical experiments are carried out to verify the validity of our theoretical analysis.  It is shown that the superconvergence orders of the elements mentioned above on anisotropic meshes are indeed as same as that of isotropic elements.

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Title: A Preserving-symmetry Finite Volume Scheme and Superconvergence on Quadrangle Grids

Abstract: Aiming at a kind of elliptic problem, a preserving-symmetry finite volume scheme on quadrangle partition grids is established in this paper.  Furthermore we give the error asymptotic expansion of finite volume solution function under rectangle grids in l²-norm.  By this expansion and combining the finite volume solution, the superconvergence result of derivative (O(h²)) is obtained in l²-norm.  Some numerical experiments verified the correctness of theoretic results.

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Title: A High Order Parallel Method for Time Discretization of Parabolic Type Equations Based on Laplace Transformation and Quadrature

Abstract: We consider the discretization in time of a parabolic equation, using a representation of the solution as an integral along a smooth curve in the complex left half plane.  The integral is then evaluated to high accuracy by a quadrature rule.  This reduces the problem to a finite set of elliptic equations, which may be solved in parallel.

The procedure is combined with finite element discretization in the spatial variables.  The method is also applied to some parabolic type evolution equations with memory.

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Title: Local A posteriori Error Estimates by Recovered Gradients on Nonuniform Meshes

Abstract: We start by recalling work by Hoffmann, Schatz, Wahlbin and Wittum (2001) which gives a theory of (some) locally based a posteriori error estimators.  It explains why they can be asserted to be asymptotically exact even in the absence of saturation conditions.  It also pays attention to when asymptotic exactness fails.

The 2001 theory is valid for quadratic or higher finite elements.  We explain how it can be extended to the basic piecewise linear case, Schatz and Wahlbin (2004).  The somewhat "surprising" technical answer is to consider the differences e(x)-e(y) in the error e at different points x and y.

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Title: Uniform l² Behavior for Time Discretization of a Volterra Equation with Completely Monotonic Kernel II: Convergence

Abstract: We extend our previous work on the time discretization for the solution of a Volterra equation with completely monotonic convolution kernel.  The method considered in time discretization comes from Part I, used the backward Euler and combined with order one convolution quadrature approximation the integral.  In present paper, the convergence properties of the discretization in time are given in the l²(0, ∞, H) norm.

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Title: An Improved Search-extension Method for Solving Multiple Solutions of Semi-linear PDEs

Abstract: Most numerical method for calculating the saddle-point solutions of semilinear elliptic are based on various minimax theorems.  In the cases, only those with "nice properties" can be numerically approximated.  Our work is geared to combine the finite element method, the interpolated coefficient finite element method (ICFEM), the eigenfunction expansion method and the search-extension method proposed by C.M. Chen and Z.Q. Xie to obtain the multiple solutions for semilinear elliptic function.  This strategy not only reduces the expensive computation greatly, but also successfully obtains multiple solutions for a class of semilinear elliptic boundary value problems with odd or non-odd nonlinearity on some symmetric and nonsymmetric domains.  Furthermore, a superconvergence result for the quadratic rectangular and triangular ICFEM on uniform meshes shall be obtained.  Numerical solutions which are illustrated by their graphic for visualization will show the efficiency of our approach.

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Title: Superconvergence Analysis for Distributed Convex Optimal Control Problems Governed by Stokes Equations

Abstract: In this talk, we present the superconvergence analysis for the finite element approximation of the distributed convex optimal control problems governed by Stokes equations.  It can be proved that if the solution is smooth and the mesh is uniform, the superconvergent error order can be proved, which is one order higher than the standard error estimate.  Based on the superconvergence analysis, the recovery type a posteriori error estimator can be constructed.  It provides the equivalent upper and lower error bound under the general conditions.  If the strong conditions are satisfied, it is not only equivalent, but also asymptotically exact.  Such estimates, which are apparently not available in the literature, can be used to construct adaptive finite element approximation schemes and as an error bound in reliability analysis for the control problems governed by Stokes equations.

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Title: A Posteriori Error Estimator for Spectral Approximations of Completely Continuous Operators

Abstract: For the spectral approximations of completely continuous operators, in order to estimate the error of
approximation eigenvalue μ_h obtained by projection method.  We have the theory depends on α, the ascent of T - μ_h.  In
this paper, the author gives a new error estimate pattern, which only depends on the ascent of T_h - μ_h, namely l (l ≤ α) which can be calculated.  Applying this estimate pattern to the orthogonal projection method of integral operator eigenvalue problem, through calculating an integration, we can get asymptotically exact indicator. And using numerical experiment we proved partial theoretical conclusions.

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Title: Higher Order Numerical Methods for Maxwell's Equations in Dispersive Media

Abstract: The discontinuous Galerkin method is applied to solve time domain Maxwell's equations in lossy and linear dispersive media.  Based on auxiliary differential equation method for dealing with dispersion, we obtain a unified formulation of Maxwell's equations in both physical dispersion region and the perfectly matched layer (PML) region.  The numerical results validate the numerical convergence and the higher order accuracy of our methods.  We also give the numerical result of the scattering wave of the buried objects in the lossy dispersive soil detected by using ground penetrating radar (GPR).

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Title: Sharp L²-Error Estimates and Superconvergence of Mixed Finite Element Methods for NonFickian Flows in Porous Media 

Abstract: A sharper L²-error estimate is obtained for the nonFickian flow of fluid in porous media by means of mixed Ritz-Volterra projection instead of the mixed Ritz projection used in [R.E. Ewing, Y. Lin, and J. Wang, Acta Math. Univ. Comenian (N.S.), 70 (2001), pp. 75-84].  Moreover, local L² superconvergence for the velocity along the Gauss lines and for the pressure at the Gauss points is derived for the mixed finite element method via the Ritz-Volterra projection, and global L² superconvergence for the velocity and the pressure is also investigated by virtue of an interpolation postprocessing technique.  On the basis of the superconvergence estimates, some useful a posteriori error estimators are presented for this mixed finite element method.

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Title: A Highly Accurate Derivative Recovery Formula for Finite Element Approximations in One Space Dimension

Abstract: A highly accurate derivative recovery formula is proposed for the k-order finite element approximations to the two-point boundary value problems.  This formula possesses the O(h^k+1) order of superconvergence on the whole domain in L_∞ norm and O(h^2k) order of ultraconvergence at the mesh points, and also the lowest regularity requirement for the exact solutions.  Numerical experiments are given to verify the high accuracy of our formula.

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Title: Superconvergence by Post-processing

Abstract: Without post-processing, superconvergence (of the finite element method) happens only at some isolated special points and depends heavily on mesh geometry.  However, with some properly designed post-processing techniques, superconvergence may valid for a whole local region, even the whole solution domain.  Furthermore, by post-processing, the mesh dependent can be relaxed significantly.

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