On the asymptotic exactness of error estimators for linear triangular finite elements

Summary This paper deals with a-posteriori error estimates for piecewise linear finite element approximations of elliptic problems. We analyze two estimators based on recovery operators for the gradient of the approximate solution. By using superconvergence results we prove their asymptotic exactness under regularity assumptions on the mesh and the solution.One of the estimators can be easily computed in terms of the jumps of the gradient of the finite element approximation. This estimator is equivalent to the error in the energy norm under rather general conditions. However, we show that for the asymptotic exactness, the regularity assumption on the mesh is not merely technical. While doing this, we analyze the relation between superconvergence and asymptotic exactness for some particular examples.     

The backward euler anisotropic a posteriori error analysis for parabolic integro‐differential equations

We derive two optimal a posteriori error estimators for an implicit fully discrete approximation to the solutions of linear integro‐differential equations of the parabolic type. A continuous, piecewise linear finite element space is used for the space discretization and the time discretization is based on an implicit backward Euler method. The a posteriori error indicator corresponding to space discretization is derived using the anisotropic interpolation estimates in conjunction with a Zienkiewicz‐Zhu error estimator to approach the error gradient. The error due to time discretization is derived using continuous, piecewise linear polynomial in time. We use the linear approximation of the Volterra integral term to estimate the quadrature error in the second estimator. Numerical experiments are performed on the isotropic mesh to validate the derived results.© 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1309–1330, 2016     

Graded mesh refinement and error estimates of higher order for DGFE solutions of elliptic boundary value problems in polygons

Error estimates for DGFE solutions are well investigated if one assumes that the exact solution is sufficiently regular. In this article, we consider a Dirichlet and a mixed boundary value problem for a linear elliptic equation in a polygon. It is well known that the first derivatives of the solutions develop singularities near reentrant corner points or points where the boundary conditions change. On the basis of the regularity results formulated in Sobolev–Slobodetskii spaces and weighted spaces of Kondratiev type, we prove error estimates of higher order for DGFE solutions using a suitable graded mesh refinement near boundary singular points. The main tools are as follows: regularity investigation for the exact solution relying on general results for elliptic boundary value problems, error analysis for the interpolation in Sobolev–Slobodetskii spaces, and error estimates for DGFE solutions on special graded refined meshes combined with estimates in weighted Sobolev spaces. Our main result is that there exist a local grading of the mesh and a piecewise interpolation by polynoms of higher degree such that we will get the same order O (h^α) of approximation as in the smooth case. © 2011 Wiley Periodicals, Inc. Numer Mehods Partial Differential Eq, 2012     

Modified cross-grid finite elements for the stokes problem

The stability of modified cross-grid elements for the approximation of the Stokes problem using continuous piecewise linear polynomials to approximate velocities and piecewise constants to approximate pressures is proved. A key feature of the method is that the mesh for pressure is modified so that the method is stable without augmenting pressure jumps. A numerical test which confirms the stability and the optimal order error estimate is presented.     

On optimal global error bounds obtained by scaled local error estimates

Summary In the second section a general method of obtaining optimal global error bounds by scaling local error estimates is developed. It is reduced to the solution of a fixpoint problem. In Sect. 3 we show more concrete error estimates reflecting a singularity of order . It is shown that under general circumstances an optimal global error bound is achieved by an (asymptotically) geometric mesh for the local error estimates. In the fourth section we specialize this to the best approximation ofg(x)x by piecewise polynomials with variable knots and degrees having a total numberN of parameters. This generalizes the result of R. DeVore and the author forg(x)=1. In the last section this problem is studied for the functione ^–x on (0, ). The exact asymptotic behaviour of the approximation withN parameters is determined toe ^qoN , whereq _o=0.895486 ....     

Expansion over a system of shifts of pyramidal functions with a square base

A method for approximation of functions of two variables by a linear combination of non-negative piecewise linear functions with a compact support is presented. Two quadratic pyramids are used as generating functions for the system of shifts. The accuracy of this local method is proved to have the same order as the best approximation by piecewise linear functions.     

Weighted inequalities and applications to best local approximation in Luxemburg norm

In this paper we establish a result about uniformly equivalent norms and the convergence of best approximant pairs on the unitary ball for a family of weighted Luxemburg norms with normalized weight functions depending on ε, when ε→ 0. It is introduced a general concept of Pade approximant and we study its relation with the best local quasi-rational approximant. We characterize the limit of the error for polynomial approximation. We also obtain a new condition over a weight function in order to obtain inequalities in Lp norm, which play an important role in problems of weighted best local Lp approximation in several variables.     

Small data oscillation implies the saturation assumption

Summary. The saturation assumption asserts that the best approximation error in with piecewise quadratic finite elements is strictly smaller than that of piecewise linear finite elements. We establish a link between this assumption and the oscillation of , and prove that small oscillation relative to the best error with piecewise linears implies the saturation assumption. We also show that this condition is necessary, and asymptotically valid provided . Received November 17, 2000 / Published online July 25, 2001     

Local discontinuous galerkin approximation of convection‐dominated diffusion optimal control problems with control constraints

In this article, we investigate local discontinuous Galerkin approximation of stationary convection‐dominated diffusion optimal control problems with distributed control constraints. The state variable and adjoint state variable are approximated by piecewise linear polynomials without continuity requirement, whereas the control variable is discretized by variational discretization concept. The discrete first‐order optimality condition is derived. We show that optimization and discretization are commutative for the local discontinuous Galerkin approximation. Because the solutions to convection‐dominated diffusion equations often admit interior or boundary layers, residual type a posteriori error estimate in L² norm is proved, which can be used to guide mesh refinement. Finally, numerical examples are presented to illustrate the theoretical findings. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 339–360, 2014     

The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges

This paper is concerned with a specific finite element strategy for solving elliptic boundary value problems in domains with corners and edges. First, the anisotropic singular behaviour of the solution is described. Then the finite element method with anisotropic, graded meshes and piecewise linear shape functions is investigated for such problems; the schemes exhibit optimal convergence rates with decreasing mesh size. For the proof, new local interpolation error estimates for functions from anisotropically weighted spaces are derived. Finally, a numerical experiment is described, that shows a good agreement of the calculated approximation orders with the theoretically predicted ones. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Analysis of an algorithm for generating locally optimal meshes for approximation by discontinuous piecewise polynomials

This paper discusses the problem of constructing a locally optimal mesh for the best approximation of a given function by discontinuous piecewise polynomials. In the one-dimensional case, it is shown that, under certain assumptions on the approximated function, Baines' algorithm [M.J. Baines, Math. Comp., 62 (1994), pp. 645-669] for piecewise linear or piecewise constant polynomials produces a mesh sequence which converges to an optimal mesh. The rate of convergence is investigated. A two-dimensional modification of this algorithm is proposed in which both the nodes and the connection between the nodes are self-adjusting. Numerical results in one and two dimensions are presented.

     

Error bounds for the finite element approximation of an incompressible material in an unbounded domain

Summary. In this paper we design high-order local artificial boundary conditions and present error bounds for the finite element approximation of an incompressible elastic material in an unbounded domain. The finite element approximation is formulated in a bounded computational domain using a nonlocal approximate artificial boundary condition or a local one. In fact there are a family of nonlocal approximate artificial boundary conditions with increasing accuracy (and computational cost) and a family of local ones for a given artificial boundary. Our error bounds indicate how the errors of the finite element approximations depend on the mesh size, the terms used in the approximate artificial boundary condition and the location of the artificial boundary. Numerical examples of an incompressible elastic material outside a circle in the plane is presented. Numerical results demonstrate the performance of our error bounds. Received August 31, 1998 / Revised version received November 6, 2001 / Published online March 8, 2002     

The sdfem on Shishkin meshes for linear convection-diffusion problems

Summary. We consider singularly perturbed linear elliptic problems in two dimensions. The solutions of such problems typically exhibit layers and are difficult to solve numerically. The streamline diffusion finite element method (SDFEM) has been proved to produce accurate solutions away from any layers on uniform meshes, but fails to compute the boundary layers precisely. Our modified SDFEM is implemented with piecewise linear functions on a Shishkin mesh that resolves boundary layers, and we prove that it yields an accurate approximation of the solution both inside and outside these layers. The analysis is complicated by the severe nonuniformity of the mesh. We give local error estimates that hold true uniformly in the perturbation parameter , provided only that , where mesh points are used. Numerical experiments support these theoretical results. Received February 19, 1999 / Revised version received January 27, 2000 / Published online August 2, 2000     

Error estimates for the finite element approximation of linear elastic equations in an unbounded domain

In this paper we present error estimates for the finite element approximation of linear elastic equations in an unbounded domain. The finite element approximation is formulated on a bounded computational domain using a nonlocal approximate artificial boundary condition or a local one. In fact there are a family of nonlocal approximate boundary conditions with increasing accuracy (and computational cost) and a family of local ones for a given artificial boundary. Our error estimates show how the errors of the finite element approximations depend on the mesh size, the terms used in the approximate artificial boundary condition, and the location of the artificial boundary. A numerical example for Navier equations outside a circle in the plane is presented. Numerical results demonstrate the performance of our error estimates.

     

Inverses for a class of banded matrices and applications to piecewise cubic approximation

We give a simple criterion for the invertibility of a class of banded matrices that arise in the approximation by piecewise cubic polynomials. We also give a formula for the inverse in terms of the powers of a 2 × 2 matrix. We present sample applications of these results to interpolation and eigenvalue problems. As a side result wee find that the Gaussian points are best.     

An adaptive strategy for elliptic problems including a posteriori controlled boundary approximation

We derive a posteriori error estimates for the approximation of linear elliptic problems on domains with piecewise smooth boundary. The numerical solution is assumed to be defined on a Finite Element mesh, whose boundary vertices are located on the boundary of the continuous problem. No assumption is made on a geometrically fitting shape.

A posteriori error estimates are given in the energy norm and the -norm, and efficiency of the adaptive algorithm is proved in the case of a saturated boundary approximation. Furthermore, a strategy is presented to compute the effect of the non-discretized part of the domain on the error starting from a coarse mesh. This especially implies that parts of the domain, where the measured error is small, stay non-discretized. The presented algorithm includes a stable path following to supply a sufficient polygonal approximation of the boundary, the reliable computation of the a posteriori estimates and a mesh adaptation based on Delaunay techniques. Numerical examples illustrate that errors outside the initial discretization will be detected.

     

Construction of basis functions in the finite element method

In many applications of the finite element method, the explicit form of the basis functions is not known. A well-known exception is that of piecewise linear approximation over a triangulation of the plane, where the basis functions are pyramid functions. In the present paper, the basis functions are displayed in closed form for piecewise polynomial approximation of degreen over a triangulation of the plane. These basis functions are expressed simply in terms of the pyramid functions for linear approximation.     

Optimally accurate Petrov-Galerkin method of finite elements

We perform analysis for a finite elements method applied to the singular self-adjoint problem. This method uses continuous piecewise polynomial spaces for the trial and the test spaces. We fit the trial polynomial space by piecewise exponentials and we apply so exponentially fitted Galerkin method to singular self-adjoint problem by approximating driving terms by Lagrange piecewise polynomials, linear, quadratic and cubic. We measure the erroe in max norm. We show that method is optimal of the first order in the error estimate. We also give numerical results for the Galerkin approximation.     

Polynomial approximation of piecewise analytic functions on a compact subset of the real line

We discuss Totik’s extension of the classical Bernstein theorem on polynomial approximation of piecewise analytic functions on a closed interval. The error of the best uniform approximation of such functions on a compact subset of the real line is studied.     

Approximating probability density functions and their convolutions using orthogonal polynomials

This paper describes and tests methods for piecewise polynomial approximation of probability density functions using orthogonal polynomials. Empirical tests indicate that the procedure described in this paper can provide very accurate estimates of probabilities and means when the probability density function cannot be integrated in closed form. Furthermore, the procedure lends itself to approximating convolutions of probability densities. Such approximations are useful in project management, inventory modeling, and reliability calculations, to name a few applications. In these applications, decision makers desire an approximation method that is robust rather than customized. Also, for these applications the most appropriate criterion for accuracy is the average percent error over the support of the density function as opposed to the conventional average absolute error or average squared error. In this paper, we develop methods for using five well-known orthogonal polynomials for approximating density functions and recommend one of them as giving the best performance overall.