Error identities for variational problems with obstacles

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F60076658%3A12310%2F18%3A43897558" target="_blank" >RIV/60076658:12310/18:43897558 - isvavai.cz</a>

Nalezeny alternativní kódy

RIV/67985556:_____/18:00483574

Výsledek na webu

<a href="https://onlinelibrary.wiley.com/doi/epdf/10.1002/zamm.201700105" target="_blank" >https://onlinelibrary.wiley.com/doi/epdf/10.1002/zamm.201700105</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1002/zamm.201700105" target="_blank" >10.1002/zamm.201700105</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Error identities for variational problems with obstacles

Popis výsledku v původním jazyce

The paper is devoted to analysis of a class of nonlinear free boundary problems that are usually solved by variational methods based on primal, dual or primal-dual variational settings. We deduce and investigate special relations (error identities). They show that a certain nonlinear measure of the distance to the exact solution (specific for each problem) is equivalent to the respective duality gap, whose minimization is the keystone of all variational numerical methods. Therefore, the identity actually sets the measure that contains maximal quantitative information on the quality of a numerical solution available through these methods. The measure has quadratic terms generated by the linear part of the differential operator and nonlinear terms associated with the free boundary. We obtain fully computable two sided bounds of this measure and show that they provide efficient estimates of the distance between the minimizer and any function (approximation) from the corresponding energy space. Several computational examples show that for different minimization sequences the balance between the quadratic and the nonlinear terms of the overall error measure may be different and essential contribution of nonlinear terms may serve as an indicator that the free boundaries are approximated very roughly.

Název v anglickém jazyce

Error identities for variational problems with obstacles

Popis výsledku anglicky

The paper is devoted to analysis of a class of nonlinear free boundary problems that are usually solved by variational methods based on primal, dual or primal-dual variational settings. We deduce and investigate special relations (error identities). They show that a certain nonlinear measure of the distance to the exact solution (specific for each problem) is equivalent to the respective duality gap, whose minimization is the keystone of all variational numerical methods. Therefore, the identity actually sets the measure that contains maximal quantitative information on the quality of a numerical solution available through these methods. The measure has quadratic terms generated by the linear part of the differential operator and nonlinear terms associated with the free boundary. We obtain fully computable two sided bounds of this measure and show that they provide efficient estimates of the distance between the minimizer and any function (approximation) from the corresponding energy space. Several computational examples show that for different minimization sequences the balance between the quadratic and the nonlinear terms of the overall error measure may be different and essential contribution of nonlinear terms may serve as an indicator that the free boundaries are approximated very roughly.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

Výsledek vznikl pri realizaci vícero projektů. Více informací v záložce Projekty.

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2018

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

ZAMM-Zeitschrift fur Angewandte Mathematik und Mechanik

ISSN

0044-2267

e-ISSN

—

Svazek periodika

98

Číslo periodika v rámci svazku

4

Stát vydavatele periodika

DE - Spolková republika Německo

Počet stran výsledku

24

Strana od-do

635-658

Kód UT WoS článku

000430013600010

EID výsledku v databázi Scopus

2-s2.0-85041107421