A posteriori error estimates for approximate solutions to variational problems with strongly convex functionals. (Russian) Zbl 0941.65059

Repin, S. I.

Probl. Mat. Anal. 17, 199-226 (1997).

Many problems involving nonlinear partial differential operators are reduced to variational problems, i.e. to finding a minimizer of some functional \(J(\omega)\). In the article under review, the functional \(J(\omega)\) is of the form \(J(\omega)=F(\omega)+G(\Lambda\omega)\), where \(F\) is a convex functional, \(G(\Lambda\omega)\) is a strongly convex functional, and \(\Lambda\) is a bounded linear operator. The functionals and the operator \(\Lambda\) are defined in some Banach space. The main aim of the article is to obtain estimates for the deviation of a solution and an approximate solution to this problem.
When constructing these error bounds, the author uses the data of the problem and an approximate solution itself and applies the duality approach in which the initial variational problem and the dual variational problem are analyzed simultaneously. The general form of a posteriori estimates is described. The case of a linear functional \(F\) is considered separately. A posteriori estimates based on approximate solutions to the dual problem are also given. The results are applied to the simplest nonlinear elliptic problem.

Reviewer: S.G.Pyatkov (Novosibirsk)

MSC:

65J15	Numerical solutions to equations with nonlinear operators
65N15	Error bounds for boundary value problems involving PDEs
47J20	Variational and other types of inequalities involving nonlinear operators (general)
35J65	Nonlinear boundary value problems for linear elliptic equations

Keywords:

convex functional; boundary value problem; error bound; variational approach; Banach space; nonlinear elliptic problem