On the Reliability of Error Indication Methods for Problems with Uncertain Data

This paper is concerned with studying the effects of uncertain data in the context of error indicators, which are often used in mesh adaptive numerical methods. We consider the diffusion equation and assume that the coefficients of the diffusion matrix are known not exactly, but within some margins (intervals). Our goal is to study the relationship between the magnitude of uncertainty and reliability of different error indication methods. Our results show that even small values of uncertainty may seriously affect the performance of all error indicators.

Two-Sided Estimates of the Solution Set for the Reaction–Diffusion Problem with Uncertain Data

We consider linear reaction–diffusion problems with mixed Dirichlet–Neumann–Robin conditions. The diffusion matrix, reaction coefficient, and the coefficient in the Robin boundary condition are defined with an uncertainty which allow bounded variations around some given mean values. A solution to such a problem cannot be exactly determined (it is a function in the set of “possible solutions” formed by generalized solutions related to possible data). The problem is to find parameters of this set. In this paper, we show that computable lower and upper bounds of the diameter (or radius) of the set can be expressed throughout problem data and parameters that regulate the indeterminacy range. Ou…

Errors Arising in Computer Simulation Methods

The goal of this introductory chapter is to discuss in general terms different classes of errors arising in computer simulation methods and to direct the reader to the chapters and sections of the book where these errors are analyzed. Moreover, we describe the error estimation methodology applied in this book.

Error Estimates for a Class of Elliptic Optimal Control Problems

In this article, functional type a posteriori error estimates are presented for a certain class of optimal control problems with elliptic partial differential equation constraints. It is assumed that in the cost functional the state is measured in terms of the energy norm generated by the state equation. The functional a posteriori error estimates developed by Repin in the late 1990s are applied to estimate the cost function value from both sides without requiring the exact solution of the state equation. Moreover, a lower bound for the minimal cost functional value is derived. A meaningful error quantity coinciding with the gap between the cost functional values of an arbitrary admissible …

Upper Bound for the Approximation Error for the Kirchhoff-Love Arch Problem

In this paper, a guaranteed and computable upper bound of approximation errors for the Kirchhoff-Love arch problem is derived. In general, it belongs to the class of functional a posteriori error estimates. The derivation method uses purely functional arguments and, therefore, the estimates are valid for any conforming approximation within the energy space. The computational implementation of the upper bound is discussed and demonstrated by a numerical example.

A Unified Approach to Measuring Accuracy of Error Indicators

In this paper, we present a unified approach to error indication for elliptic boundary value problems. We introduce two different definitions of the accuracy (weak and strong) and show that various indicators result from one principal relation. In particular, this relation generates all the main types of error indicators, which have already gained high popularity in numerical practice. Also, we discuss some new forms of indicators that follow from a posteriori error majorants of the functional type and compare them with other indicators. Finally, we discuss another question related to accuracy of error indicators for problems with incompletely known data.

Worst case approach in convex minimization problems with uncertain data

This paper concerns quantitative analysis of errors generated by incompletely known data in convex minimization problems. The problems are discussed in the mixed setting and the duality gap is used as the fundamental error measure. The influence of the indeterminate data is measured using the worst case scenario approach. The worst case error is decomposed into two computable quantities, which allows the quantitative comparison between errors resulting from the inaccuracy of the approximation and the data uncertainty. The proposed approach is demonstrated on a paradigm of a nonlinear reaction-diffusion problem together with numerical examples.

Functional A Posteriori Error Estimate for a Nonsymmetric Stationary Diffusion Problem

In this paper, a posteriori error estimates of functional type for a stationary diffusion problem with nonsymmetric coefficients are derived. The estimate is guaranteed and does not depend on any particular numerical method. An algorithm for the global minimization of the error estimate with respect to an auxiliary function over some finite dimensional subspace is presented. In numerical tests, global minimization is done over the subspace generated by Raviart-Thomas elements. The improvement of the error bound due to the p-refinement of these spaces is investigated.

New Indicators of Approximation Errors for Problems in Continuum Mechanics

In this paper we present a new error indicator for approximate solutions of elliptic problems. We discuss error indication with the paradigm of the diffusion problem, however the techniques are easily adaptable to more complicated elliptic problems, for example to linear elasticity, viscous flow models and electromagnetic models. The proposed indicator does not contain mesh dependent constants and it admits parallelization. nonPeerReviewed

Guaranteed Error Bounds I

In Chap. 3, we discussed the main ideas of fully reliable error control methods and the corresponding numerical algorithms with the paradigm of simple elliptic type problems. This chapter is intended to show a deep connection between a posteriori estimates of the functional type and physical relations generating the problem. Also, the goal of this chapter is to consider a wider set of problems arising in various applications and explain things in terms of computational mechanics. For this purpose, we begin with a simple class of mechanical problems (straight beams) and after that consider curvilinear beams and more complicated models of continuum mechanics (linear elasticity, viscous fluids…

A posteriori error estimates for a Maxwell type problem

In this paper, we discuss a posteriori estimates for the Maxwell type boundary-value problem. The estimates are derived by transformations of integral identities that define the generalized solution and are valid for any conforming approximation of the exact solution. It is proved analytically and confirmed numerically that the estimates indeed provide a computable and guaranteed bound of approximation errors. Also, it is shown that the estimates imply robust error indicators that represent the distribution of local (inter-element) errors measured in terms of different norms. peerReviewed

Indicators of Errors for Approximate Solutions of Differential Equations

Error indicators play an important role in mesh-adaptive numerical algorithms, which currently dominate in mathematical and numerical modeling of various models in physics, chemistry, biology, economics, and other sciences. Their goal is to present a comparative measure of errors related to different parts of the computational domain, which could suggest a reasonable way of improving the finite dimensional space used to compute the approximate solution. An “ideal” error indicator must possess several properties: efficiency, computability, and universality. In other words, it must correctly reproduce the distribution of errors, be indeed computable, and be applicable to a wide set of approxi…

Errors Generated by Uncertain Data

In this chapter, we study effects caused by incompletely known data. In practice, the data are never known exactly, therefore the results generated by a mathematical model also have a limited accuracy. Then, the whole subject of error analysis should be treated in a different manner, and accuracy of numerical solutions should be considered within a framework of a more complicated scheme, which includes such notions as maximal and minimal distances to the solution set and its radius.

Analysis of errors caused by incomplete knowledge of material data in mathematical models of elastic media

Overview of Other Results and Open Problems

This chapter presents an overview of results related to error control methods, which were not considered in previous chapters. In the first part, we discuss possible extensions of the theory exposed in Chaps. 3 and 4 to nonconforming approximations and certain classes of nonlinear problems. Also, we shortly discuss some results related to explicit evaluation of modeling errors. The remaining part of the chapter is devoted to a posteriori estimates of errors in iteration methods. Certainly, the overview is not complete. A posteriori error estimation methods are far from having been fully explored and this subject contains many unsolved problems and open questions, some of which we formulate …