Abstract:
A posteriori estimates are proved for the accuracy of approximations of solutions to variational problems with nonstandard power functionals. More precisely, these are integral functionals with power type integrands having a variable exponent p(⋅). It is assumed that p(⋅) is bounded away from one and infinity. Estimates in the energy norm are obtained for the difference of the approximate and exact solutions. The majorant M in these estimates depends only on the approximation v and the data of the problem, but is independent of the exact solution u. It is shown that M=M(v) vanishes as v tends to u and M(v)=0 only if v=u. The superquadratic and subquadratic cases (which means that p(⋅)⩾2, or p(⋅)⩽2, respectively) are treated separately.
Keywords:
variational problem with nonstandard coercivenes and growth conditions, a posteriori error estimates for approximate solutions, dual problem.
Citation:
S. E. Pastukhova, “A posteriori estimates of the deviation from exact solutions to variational problems under nonstandard coerciveness and growth conditions”, Algebra i Analiz, 32:1 (2020), 51–77; St. Petersburg Math. J., 32:1 (2021), 39–57
\Bibitem{Pas20}
\by S.~E.~Pastukhova
\paper A posteriori estimates of the deviation from exact solutions to variational problems under nonstandard coerciveness and growth conditions
\jour Algebra i Analiz
\yr 2020
\vol 32
\issue 1
\pages 51--77
\mathnet{http://mi.mathnet.ru/aa1682}
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\transl
\jour St. Petersburg Math. J.
\yr 2021
\vol 32
\issue 1
\pages 39--57
\crossref{https://doi.org/10.1090/spmj/1637}
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Linking options:
https://www.mathnet.ru/eng/aa1682
https://www.mathnet.ru/eng/aa/v32/i1/p51
This publication is cited in the following 1 articles:
S. I. Repin, “A posteriori identities for measures of deviation from exact solutions of nonlinear boundary value problems”, Comput. Math. Math. Phys., 63:6 (2023), 934–956