Superconvergence Result Research Articles

Two order superconvergence of finite element methods for Sobolev equations

We consider finite element methods applied to a class of Sobolev equations inRd(d ≥ 1). Global strong superconvergence, which only requires that partitions are quais-uniform, is investigated for the error between the approximate solution and the Ritz-Sobolev projection of the exact solution. Two order superconvergence results are demonstrated inW1,p(Ω) andLp(Ω) for 2 ≤p < ∞.

Local Expansions of Periodic Spline Interpolation with Some Applications

In this paper we prove some asymptotic expansions of the error of interpolation on equally spaced nodes with periodic smoothest splines of arbitrary degree on a uniform partition. We obtain a local expansion in terms of derivatives of the interpolate. Afterwards we apply this result to the asymptotic study of the numerical solution of periodic integral equations of the second kind by means of ϵ – collocation methods. We show some new superconvergence results and give particular forms of these expansions depending on the choices of the parameter ϵ. We finally give some numerical experiments, which corroborate the theory.

Superconvergence of Finite Element Approximations for the Stokes Problem by Projection Methods

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.

Superconvergence of the Local Discontinuous Galerkin Method for Elliptic Problems on Cartesian Grids

In this paper, we present a superconvergence result for the local discontinuous Galerkin (LDG) method for a model elliptic problem on Cartesian grids. We identify a special numerical flux for which the L2-norm of the gradient and the L2-norm of the potential are of orders k+1/2 and k+1, respectively, when tensor product polynomials of degree at most k are used; for arbitrary meshes, this special LDG method gives only the orders of convergence of k and k+1/2, respectively. We present a series of numerical examples which establish the sharpness of our theoretical results.

A critical study of hypersingular and strongly singular boundary integral representations of potential gradient

This paper deals with recovery of potential gradient (∇u) in the resolution of the two-dimensional Laplace equation by boundary element method. The emphasis is placed on ∇u-recovery near and on the boundary, where error behaviour and convergence rate are studied in detail. Conventional hypersingular (HS) and strongly singular (SS) boundary integral representations (BIRs) of ∇u are compared with two recovery procedures recently introduced by Mantic, Graciani and Paris, Comput. Methods Appl. Mech. Engrg (1999) 178, pp. 267–289: DSC – a local smoothing procedure, and SSC - applying SS BIR with the integral density obtained previously from DSC. The theoretical analysis of error of ∇u recovered by ∇u-BIRs presented is supported by numerical examples. The numerical study of convergence carried out using an h-refinement of (quasi)uniform meshes of linear boundary elements shows a slow convergence rate, O(h) or less, of HS and SS BIRs in comparison with superconvergence rate O(h2) of DSC (only on the boundary) and SSC near the boundary and at element centres and junctions between elements. A slow convergence rate is obtained by all approximations at boundary corners, SSC clearly providing the most accurate results. Superconvergent results of HS and SS BIRs are only obtained in a particular case of evaluation of tangential derivative of potential on a straight part of the boundary.

The numerical solution of integral-algebraic equations of index 1 by polynomial spline collocation methods

In this paper, we study polynomial spline collocation methods applied to a particular class of integral-algebraic equations of Volterra type. We analyse mixed systems of second and first kind integral equations. Global convergence and local superconvergence results are established.

Improving the convergence of some boundary element methods

We will discuss three possibilities of how to improve the approximation quality of boundary element approximations for integral equations of the first kind on smooth planar curves. First we comment on iterated finite element solutions, then on reduced integration techniques resulting in so-called qualocation methods. As a third option we present post-processing based on a recent superconvergence result of the author. For this last approach we present numerical experiments in the context of adaptive refinement.

A comparison of a posteriori error estimators for mixed finite element discretizations by Raviart-Thomas elements

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.

Superconvergence of Mixed Finite Element Approximations over Quadrilaterals

A superconvergence result is established in this article for approximate solutions of second-order elliptic equations by mixed finite element methods over quadrilaterals. The superconvergence indicates an accuracy of ${\cal O}(h^{k+2})$ for the mixed finite element approximation if the Raviart--Thomas or Brezzi--Douglas--Fortin--Marini elements of order k are employed with optimal error estimate of ${\cal O}(h^{k+1})$. Numerical experiments are presented to illustrate the theoretical result.

On the supraconvergence of elliptic finite difference schemes

This paper deals with the supraconvergence of elliptic finite difference schemes on variable grids for second order elliptic boundary value problems subject to Dirichlet boundary conditions in two-dimensional domains. The assumptions in this paper are less restrictive than those considered so far in the literature allowing also variable coefficients, mixed derivatives and polygonal domains. The nonequidistant grids we consider are more flexible than merely rectangular ones such that, e.g., local grid refinements are covered. The results also develop a close relation between supraconvergent finite difference schemes and piecewise linear finite element methods. It turns out that the finite difference equation is a certain nonstandard finite element scheme on triangular girds combined with a special form of quadrature. In extension to what is known for the standard finite element scheme, here also the gradient is shown to be convergent of second order, and so our result is also a superconvergence result for the underlying finite element method.

The Collocation Method for Solving the Radiosity Equation for Unoccluded Surfaces

The radiosity equation occurs in computer graphics, and its solution leads to more realistic illumination for the display of surfaces. We consider the behavior of the radiosity integral operator, for smooth and piecewise smooth surfaces. A collocation method for solving the radiosity equation is proposed and analyzed. The method uses piecewise linear interpolation; and for one particular choice of such linear interpolation, it is shown that superconvergence results are obtained when solving on a smooth surface. Numerical results conclude the paper.

Superconvergence in the projected‐shear plate‐bending finite element method

A projected-shear finite element method for periodic Reissner–Mindlin plate model are analyzed for rectangular meshes. A projection operator is applied to the shear stress term in the bilinear form. Optimal error estimates in the L2-norm, the H1-norm, and the energy norm for both displacement and rotations are established and gradient superconvergence along the Gauss lines is justified in some weak senses. All the convergence and superconvergence results are uniform with respect to the thickness parameter t. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 367–386, 1998

Superconvergence in the projected-shear plate-bending finite element method

A projected-shear finite element method for periodic Reissner–Mindlin plate model are analyzed for rectangular meshes. A projection operator is applied to the shear stress term in the bilinear form. Optimal error estimates in the L2-norm, the H1-norm, and the energy norm for both displacement and rotations are established and gradient superconvergence along the Gauss lines is justified in some weak senses. All the convergence and superconvergence results are uniform with respect to the thickness parameter t. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 367–386, 1998

Multi-parameter error resolution for the collocation method of Volterra integral equations

In this paper, a multi-parameter error resolution technique is introduced and applied to the collocation method for Volterra integral equations. By using this technique, an approximation of higher accuracy is obtained by using a multi-processor in parallel. Additionally, a correction scheme for approximation of higher accuracy and a global superconvergence result are presented.

A note on the convergence of product integration and Galerkin method for weakly singular integral equations

We consider weakly singular integral equations of Fredholm-type whose kernels satisfy certain algebraic estimates with their derivatives. In particular, we establish optimal convergence order estimates for product integration and Galerkin method applied on suitable grading mesh for the solution of such equations. Some superconvergence results are also derived.

Adaptive Multilevel Techniques for Mixed Finite Element Discretizations of Elliptic Boundary Value Problems

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. By a well-known postprocessing technique the discrete problem is equivalent to a modified nonconforming discretizationwhich is solved by preconditioned CG iterations using a multilevel preconditioner in the spirit of Bramble, Pasciak, and Xu designed for standard nonconforming approximations. Local refinement of the triangulations is based on an a posteriori error estimator which can be easily derived from superconvergence results. The performance of the preconditioner and the error estimator is illustrated by several numerical examples.

Convergence estimates for the numerical approximation of homoclinic solutions

This article is concerned with the numerical computation of homoclinic solutions converging to a hyperbolic or semi-hyperbolic equilibrium of a system u = f(u, μ). The approximation is done by replacing the original problem with a boundary value problem on a finite interval and introducing an additional phase condition to make the solution unique. Numerical experiments have indicated that the parameter μ is much better approximated than the homoclinic solution. This was proved in Schecter (1995 IMA J. Numer Anal. 15, 23–60) for phase conditions satisfying an additional ‘niceness’ assumption, which is unfortunately not satisfied for the phase condition most commonly used in numerical experiments and which actually suggested the super-convergence result. Here, this result is proved for arbitrary phase conditions. Moreover, it is shown that it suffices to approximate the original boundary value problem to first order when considering semi-hyperbolic equilibria, extending a result of Schecter (1993 SIAM J. Numer Anal. 30, 1155–78).

Superconvergence of the iterated collocation methods for Hammerstein equations

In this paper, we analyse the iterated collocation method for Hammerstein equations with smooth and weakly singular kernels. The paper expands the study which began in [16] concerning the superconvergence of the iterated Galerkin method for Hammerstein equations. We obtain in this paper a similar superconvergence result for the iterated collocation method for Hammerstein equations. We also discuss the discrete collocation method for weakly singular Hammerstein equations. Some discrete collocation methods for Hammerstein equations with smooth kernels were given previously in [3, 18].

On Superconvergence Results and Negative Norm Estimates for Parabolic Integro-Differential Equations

The purpose of this paper is to show howknown negative norm estimates and superconvergence resultsapplied to parabolic equations can be carried over to integro-differential equations of parabolic type. A quasi projectiontechnique introduced earlier by Douglas, Dupont and Wheeleris modified to establish negative norm estimates in severalspace variables. Further, in a single space variable, knot su-perconvergence is also established. Finally, interior supercon-vergence estimates are also derived.

Stieltjes Derivatives and $\beta $-Polynomial Spline Collocation for Volterra Integrodifferential Equations with Singularities

In this paper, we introduce a new tool, the Stieltjes Derivative, and new kinds of meshes which are more practical than the graded meshes. Based on these, we discuss a nonpolynomial spline collocation method for Volterra integrodifferential equations with weakly singular kernels. Under very weak hypotheses which allow the nonsmooth behaviour of the given functions in these equations, collocation on such meshes is shown to yield optimal convergence rates. An ideal super-convergence result is also obtained.