A posteriori error estimation for variational problems with uniformly convex functionals

Abstract

Abstract. The objective of this paper is to introduce a general scheme for deriving a posteriori error estimates by using duality theory of the calculus of variations. We consider variational problems of the formwhere F : V → R is a convex lower semicontinuous functional, G : Y → R is a uniformly convex functional, V and Y are reflexive Banach spaces, and Λ : V → Y is a bounded linear operator. We show that the main classes of a posteriori error estimates known in the literature follow from the duality error estimate obtained and, thus, can be justified via the duality theory.

Estimates of dimension reduction errors for stationary reaction–diffusion problems

“…Our analysis of modeling errors is based upon the so-called functional-type a posteriori error estimates (see [9] - [15] and the references cited therein). More precisely, we use such type estimate derived for the reaction-diffusion equation with mixed Dirichlet-Robin boundary conditions (see [11], estimates (4.2.21) and (4.2.22)).…”

A Posteriori Estimation of Dimension Reduction Errors

“…In order to control the dimension reduction error, we apply the functional-type a posteriori error estimate derived in [8] (see also [5] and [7]) to the original three-dimensional problem (2.8). The estimate reads as follows: For all γ > 0, δ > 0 and y * ∈ H * (Ω, Div) there holds…”

“…At the same time, this method forms a basis for the hierarchical modelling of three-dimensional plates (see, e.g., [11], [3], [10]). We advocate the functional-type a posteriori error estimation approach (see [5], [6], [7], [8]) that essentially differs from the approaches taken in the aforementioned articles; however, surprisingly enough, it is possible to show that Babuška and Schwab's estimator for the zero-order reduced problem can be obtained as a particular case of our estimator when the right-hand side of the equation is zero and the original domain is a plate with plane parallel faces. It must be also noticed that the treatment of the case with non-zero right-hand side may require a special care, as we are about to see in one of the numerical examples; the presented estimator exhibits sufficient flexibility to remain efficient in this case.…”

A Posteriori Estimates for Cost Functionals of Optimal Control Problems

“…Letỹ be any function from the admissible set Y := H 1 0 (Ω), which we view as an approximation of the solution of the elliptic problem (2a)-(2b). It was shown (see, e.g., [11] and [12]) that the error of the approximationỹ satisfies the following estimate:…”