Olli Mali

A posteriori error estimates for a Maxwell type problem

Abstract 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.

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. Our method is based on using a posteriori error majorants and minorants of the functional type (see [5, 6]), which explicitly depend on the coefficients and allow to obtain the corresponding lower and upper bounds by solving the respective extremal problems generated by indeterminacy of coefficients.

Conforming and non-conforming functional a posteriori error estimates for elliptic boundary value problems in exterior domains: theory and numerical tests

Abstract - This paper is concerned with the derivation of conforming and non-conforming functional a posteriori error estimates for elliptic boundary value problems in exterior domains. These estimates provide computable and guaranteed upper and lower bounds for the difference between the exact and the approximate solution of the respective problem. We extend the results from [5] to non-conforming approximations, which might not belong to the energy space and are just considered to be square integrable. Moreover, we present some numerical tests.

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.

Blowup of errors caused by inexact knowledge of the Poisson ratio in some elasticity problems

Abstract We study the effects caused by inexact knowledge of the Poissons ratio in the linear elasticity problem and derive estimates for the incremental quantity that characterizes the variability of the internal energy and the logarithmic derivative of the energy with respect to the Poissons ratio. These quantities characterize errors caused by inexact (incomplete) knowledge of material constants, which is typical for computer modelling of physical and engineering problems.We prove that their behaviour is drastically different for two different classes of boundary conditions, namely the boundary conditions of the first class generate solutions that are relatively stable with respect to small variations of the Poissons ratio, while other conditions generate solutions, for which the quantities blow up if the ratio tends to one half.

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.

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 in the last section.

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.

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.

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.