Reliable control over approximation errors by functional type a posteriori estimates

Abstract

FM Maxim Frolovin tieteellisen laskennan väitöskirjan ”Reliable Control over Approximation Errors by Functional Type A Posteriori Estimates” tarkastustilaisuus. Vastaväittäjänä professori Rolf Stenberg (Teknillinen korkeakoulu) ja kustoksena professori Pekka Neittaanmäki.Reaalimaailman ilmiöiden matemaattisessa ja numeerisessa mallittamisessa on keskeinen kysymys, kuinka tarkka tietokoneella toteutettava numeerinen malli on. Frolovin työssä esitetään uusi menetelmä, jolla numeerisen simulointimallin tarkkuutta voidaan laskennallisesti arvioida. Sekä kehitetyn uuden menetelmän teoria että käytännön testaus on toteutettu neljännen kertaluvun lineaarisen osittaisdifferentiaaliyhtälön tapauksessa.

This thesis is focused on the development and numerical justification of a modern computational methodology that provides guaranteed upper bounds of the energy error norms. The methodology considered is based on the so-called functional type a posteriori error estimates, which have been recently suggested for problems that can be represented as problems of minimization of convex functionals. For boundary-value problems arising in the theory of plates, several new a posteriori error estimates (Duality Error Majorants) are derived on purely functional grounds either by the methods of duality theory in the calculus of variations or by modifying respective integral identities. This important feature makes it possible to take into account not only ”pure” approximation errors, but also all other errors contained in the approximate solution (including also the errors caused by possible defects of a computer code).Numerical tests are performed for elliptic type boundary-value problems of the second and fourth order. In particular, the method is compared with some other (classical) error indicators and estimators, which are based on gradient recovery or an explicit estimation of residuals. It is shown that functional type a posteriori error estimates provide accurate and reliable upper bounds of the energy norm ofthe actual error and, also, indicate elements with relatively large local errors.