Pointwise error estimates for interpolation

Pointwise error estimates are obtained for polynomial interpolants in the roots and extrema of the Chebyshev polynomials of the first kind. These estimates are analogous to those derived by Henrici [2] for trigonometric polynomial interpolants.     

Pointwise error estimates for discontinuous Galerkin methods with lifting operators for elliptic problems

In this article, we prove some weighted pointwise estimates for three discontinuous Galerkin methods with lifting operators appearing in their corresponding bilinear forms. We consider a Dirichlet problem with a general second-order elliptic operator.

     

Pointwise error estimates of the local discontinuous Galerkin method for a second order elliptic problem

In this paper we derive some pointwise error estimates for the local discontinuous Galerkin (LDG) method for solving second-order elliptic problems in (). Our results show that the pointwise errors of both the vector and scalar approximations of the LDG method are of the same order as those obtained in the norm except for a logarithmic factor when the piecewise linear functions are used in the finite element spaces. Moreover, due to the weighted norms in the bounds, these pointwise error estimates indicate that when at least piecewise quadratic polynomials are used in the finite element spaces, the errors at any point depend very weakly on the true solution and its derivatives in the regions far away from . These localized error estimates are similar to those obtained for the standard conforming finite element method.

     

Pointwise error estimates for trigonometric interpolation and conjugation

An estimate due to Gaier [2] for the error committed in replacing a periodic function f by an interpolating trigonometric polynomial is sharpened in such a way that the estimate makes evident the interpolating property of the polynomial. A similar improvement is given for Gaier's estimate of the difference between the conjugate of f and the conjugate trigonometric polynomial.     

Anisotropic error estimates for elliptic problems

Summary.  Moving from the anisotropic interpolation error estimates derived in [12], we provide here both a-priori and a-posteriori estimates for a generic elliptic problem. The a-priori result is deduced by following the standard finite element theory. For the a-posteriori estimate, the analysis extends to anisotropic meshes the theory presented in [3–5]. Numerical test-cases validate the derived results. Received July 22, 2001 / Revised version received March 20, 2002 / Published online July 18, 2002 Mathematics Subject Classification (1991): 65N15, 65N50     

Pointwise a posteriori error control for elliptic obstacle problems

We consider a finite element method for the elliptic obstacle problem over polyhedral domains in ^d, which enforces the unilateral constraint solely at the nodes. We derive novel optimal upper and lower a posteriori error bounds in the maximum norm irrespective of mesh fineness and the regularity of the obstacle, which is just assumed to be Hölder continuous. They exhibit optimal order and localization to the non-contact set. We illustrate these results with simulations in 2d and 3d showing the impact of localization in mesh grading within the contact set along with quasi-optimal meshes. Partially supported by NSF Grant DMS-9971450 and NSF/DAAD Grant INT-9910086.Partially suported by DAAD/NSF grant ``Projektbezogene Förderung des Wissenschaftleraustauschs in den Natur-, Ingenieur- und den Sozialwissenschaften mit der NSF'.Partially supported by DAAD/NSF grant ``Projektbezogene Förderung des Wissenschaftleraustauschs in den Natur-, Ingenieur- und den Sozialwissenschaften mit der NSF', and by the TMR network ``Viscosity solutions and their Applications', Italian M.I.U.R. projects ``Scientific Computing: Innovative Models and Numerical Methods' and ``Symmetries, Geometric Structures, Evolution and Memory in PDEs'.Mathematics Subject Classification (1991):65N15, 65N30, 35J85     

Pointwise a posteriori error estimates for monotone semi-linear equations

We derive upper and lower a posteriori estimates for the maximum norm error in finite element solutions of monotone semi-linear equations. The estimates hold for Lagrange elements of any fixed order, non-smooth nonlinearities, and take numerical integration into account. The proof hinges on constructing continuous barrier functions by correcting the discrete solution appropriately, and then applying the continuous maximum principle; no geometric mesh constraints are thus required. Numerical experiments illustrate reliability and efficiency properties of the corresponding estimators and investigate the performance of the resulting adaptive algorithms in terms of the polynomial order and quadrature.     

Pointwise localized error estimates for parabolic finite element equations

Summary. We derive pointwise weighted error estimates for a semidiscrete finite element method applied to parabolic equations. The results extend those obtained by A.H. Schatz for stationary elliptic problems. In particular, they show that the error is more localized for higher order elements. Mathematics Subject Classification (2000): 65N30     

Efficient and reliable hierarchical error estimates for an elliptic obstacle problem

We present and analyze novel hierarchical a posteriori error estimates for a self-adjoint elliptic obstacle problem. Under a suitable saturation assumption, we prove the efficiency and reliability of our hierarchical estimates. The proof is based upon some new observations on the efficiency of some hierarchical error indicators. These new observations allow us to remove an additional regularity condition on the underlying grid required in the previous analysis. Numerical computations confirm our theoretical findings.     

Pointwise error estimates for a streamline diffusion finite element scheme

Summary Pointwise error estimates for a streamline diffusion scheme for solving a model convection-dominated singularly perturbed convection-diffusion problem are given. These estimates improve pointwise error estimates obtained by Johnson et al.[5].     

Pointwise error estimates for a generalized Oseen problem and an application to an optimal control problem

We derive pointwise error estimates for a generalized Oseen when it is approximated by a low order Taylor‐Hood finite element scheme in two dimensions. The analysis is based on estimates for regularized Green's functions associated with a generalized Oseen problem on weighted Sobolev spaces and weighted interpolation results. We apply the maximum norm results to obtain convergence in an optimal control problem governed by a generalized Oseen equation and present a numerical example that allows us to show the behavior of the error.     

Pointwise error estimates for a stabilized Galerkin method: non-selfadjoint problems

Highly localized pointwise error estimates for a stabilized Galerkin method are provided for second-order non-selfadjoint elliptic partial differential equations. The estimates show a local dependence of the error on the derivative of the solution u and weak dependence on the global norm. The results in this paper are an extension of the previous pointwise error estimates for the self-adjoint problems. In order to provide pointwise error estimates in the presence of the first-order term in the differential equations, we prove that the stabilized Galerkin solution is higher order perturbation to the Ritz projection of the true solutions. Then, we proceed to obtain pointwise estimates using the so-called discrete Green’s function. Application to error expansion inequalities and a posteriori error estimators are briefly discussed.     

Pointwise estimates in a class of fourth order nonlinear elliptic equations

Summary The subharmonicity of a function, which is defined on solutions of a fourth order nonlinear elliptic differential equation, leads to estimates on the solution and the gradient and Laplacian of the solution at interior points in terms of values on the boundary.

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Zusammenfassung Die subharmonische Eigenschaft von Funktionen, die durch die Lösungen von nichtlinearen elliptischen Differentialgleichungen vierter Ordnung bestimmt sind, führt zu punktweiser Abschätzung der Lösung, deren Gradienten, und Laplace Operatoren, abhängig von den Randwerten sind.

     

Pointwise error estimates of the bilinear SDFEM on Shishkin meshes

A model singularly perturbed convection–diffusion problem in two space dimensions is considered. The problem is solved by a streamline diffusion finite element method (SDFEM) that uses piecewise bilinear finite elements on a Shishkin mesh. We prove that the method is convergent, independently of the diffusion parameter ε, with a pointwise accuracy of almost order 11/8 outside and inside the boundary layers. Numerical experiments support these theoretical results. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Residual type a posteriori error estimates for elliptic obstacle problems

Summary. A posteriori error estimators of residual type are derived for piecewise linear finite element approximations to elliptic obstacle problems. An instrumental ingredient is a new interpolation operator which requires minimal regularity, exhibits optimal approximation properties and preserves positivity. Both upper and lower bounds are proved and their optimality is explored with several examples. Sharp a priori bounds for the a posteriori estimators are given, and extensions of the results to double obstacle problems are briefly discussed. Received June 19, 1998 / Published online December 6, 1999     

Quantitative error estimates for coefficient idealization in linear elliptic problems

We give a thorough quantitative error analysis for the effect of coefficient idealization on solutions of linear elliptic boundary value problems. The a posteriori error estimate is derived by a tactful application of the duality theory in convex analysis. The estimate involves an auxiliary function subject to certain constraint. We discuss in detail the selection of a good auxiliary function for various cases. Numerical examples show the effectiveness of our a posteriori error estimate.     

Pointwise gradient estimates

We survey a number of recent results concerning the possibility of proving pointwise gradient estimates via potentials for solutions to quasilinear, possibly degenerate, elliptic and parabolic equations.     

Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids: Part I. Global estimates

This part contains new pointwise error estimates for the finite element method for second order elliptic boundary value problems on smooth bounded domains in . In a sense to be discussed below these sharpen known quasi-optimal and estimates for the error on irregular quasi-uniform meshes in that they indicate a more local dependence of the error at a point on the derivatives of the solution . We note that in general the higher order finite element spaces exhibit more local behavior than lower order spaces. As a consequence of these estimates new types of error expansions will be derived which are in the form of inequalities. These expansion inequalities are valid for large classes of finite elements defined on irregular grids in and have applications to superconvergence and extrapolation and a posteriori estimates. Part II of this series will contain local estimates applicable to non-smooth problems.

     

Functional a posteriori error estimates for elliptic problems in exterior domains

This paper is concerned with the derivation of computable and guaranteed upper bounds of the difference between the exact and approximate solutions of an exterior domain boundary value problem for a linear elliptic equation. Our analysis is based upon purely functional argumentation and does not attract specific properties of an approximation method. Therefore, the estimates derived in the paper at hand are applicable to any approximate solution that belongs to the corresponding energy space. Such estimates (also called error majorants of functional type) were derived earlier for problems in bounded domains of RN. Bibliography: 4 titles. Illustrations: 1 figure.