A POSTERIORI ERROR ESTIMATES FOR APPROXIMATE SOLUTIONS OF ELLIPTIC BOUNDARY VALUE PROBLEMS IN TERMS OF LOCAL NORMS AND OBJECTIVE FUNCTIONALS

Capa

Citar

Texto integral

Acesso aberto Acesso aberto
Acesso é fechado Acesso está concedido
Acesso é fechado Somente assinantes

Resumo

Functional relations have been obtained that allow us to evaluate the accuracy of approximate solutions in terms of measures significantly different from the energy norms that are usually used for these purposes. In particular, they are applicable to local norms and measures constructed using specially built linear functionals. The need for such precision control tools arises if there is a special interest in the behavior of the solution in some subdomain or in the special properties of the solution. It is shown that a posteriori functional-type estimates, which were previously used for global estimates, can be adapted to solve this problem. Functional identities and estimates are obtained that allow estimating the error of any conformal approximations in terms of a wide class of measures, including local norms and problem-oriented functionals. The theoretical results are verified in a series of examples that confirm the effectiveness of the proposed method.

Sobre autores

A. Muzalevsky

Peter the Great St. Petersburg Polytechnic University

St. Petersburg, Russia

S. Repin

St. Petersburg Department of the V.A. Steklov Mathematical Institute of the Russian Academy of Sciences

Email: repin@pdmi.ras.ru
St. Petersburg, Russia; St. Petersburg, Russia; Peter the Great St. Petersburg Polytechnic University

M. Frolov

Peter the Great St. Petersburg Polytechnic University

St. Petersburg, Russia

Bibliografia

  1. Bangerth W., Rannacher R. Adaptive finite element methods for differential equations. Berlin: Birkhauser, 2003.
  2. Johnson C., Hansbo P. Adaptive finite elements in computational mechanics // Comput. Methods Appl. Mech. Engrg. 1992. V. 101. P. 143–181.
  3. Johnson C., Szepessy A. Adaptive finite element methods for conservation laws based on a posteriori error estimates // Commun. Pure and Appl. Math. 1995. V. XLVIII. P. 199–234.
  4. Mommer M.S., Stevenson R. A goal-oriented adaptive finite element method with convergence rates // SIAM J. Numer. Anal. 2009. V. 47. № 2. P. 861—886.
  5. Stein E., Ruter M., Ohnimus S. Error-controlled adaptive goal-oriented modeling and finite element approximations in elasticity // Comput. Meth. Appl. Mech. Engrg. 2007. V. 196. № 37—40. P. 3598—3613.
  6. Repin S. I. A posteriori error estimation for variational problems with uniformly convex functionals // Math. Comput. 2000. V. 69. № 230. P. 481–500.
  7. Repin S. A posteriori estimates for partial differential equations. Berlin: Walter de Gruyter GmbH & Co. KG, 2008.
  8. Репин С.И. Тождество для отклонений от точного решения задачи A u + ` = 0 и его следствия // Ж. вычисл. матем. и матем. физ. 2021. Т. 61. № 12. С. 22–45.
  9. Repin S., Sauter S. Accuracy of Mathematical Models. Dimension Reduction, Simplification, and Homogenization. EMS Tracts in Mathematics. Vol. 33. 2020.
  10. Репин С.И. Оценки отклонения от точных решений краевых задач в мерах более сильных, чем энергетическая норма // Ж. вычисл. матем. и матем. физ. 2020. Т. 60. № 5. С. 767–783
  11. Repin S.I. A posteriori estimates in local norms // J.Math. Sci. 2004. V. 124 № 3. P. 5026–5035.
  12. Prager W., Synge J.L. Approximations in elasticity based on the concept of functions space // Quart. Appl. Math. 1947. V. 5. P. 241–269.
  13. Mikhlin S.G. Variational Methods in Mathematical Physics. Oxford: Pergamon Press, 1964.

Arquivos suplementares

Arquivos suplementares
Ação
1. JATS XML
Baixar

Declaração de direitos autorais © Russian Academy of Sciences, 2024