A Posteriori Control of Dimension Reduction Errors on Long Domains

Abstract

In this work, we consider linear elliptic problems posed in long domains, i.e. the domains whose size in one coordinate direction is much greater than the size in the other directions. If the variation of the coefficients and right-hand side along the emphasized direction is small, the original problem can be reduced to a lower-dimensional one that is supposed to be much easier to solve. The a-posteriori estimation of the error stemming from the model reduction constitutes the goal of the present work. For general coefficient matrix and right-hand side of the equation, the reliable and efficient error estimator is derived that provides a guaranteed upper bound for the modelling error, exhibits the optimal asymptotics as the size of the domain tends to infinity and correctly indicates the local error distribution.

Combineda posteriorimodeling-discretization error estimate for elliptic problems with complicated interfaces

“…Our method differs from these approaches and its derivation is based on our previous publications (see [19][20][21][22][23][24][25][26][27][28][29][30][31]), in which estimates of the difference between the exact solution of boundary value problems and arbitrary functions from the corresponding energy space has been derived by purely functional methods without requiring specific information on the approximating subspace and the numerical method used. As a result, the estimates contain no mesh dependent constants and are valid for any conforming approximation from the respective energy space.…”

“…In contrast our approach is also applicable to problems that have no variational (primal/dual) energy formulation (see [21][22][23]). Explicit and computable estimates of modeling errors related to dimension reduction models of diffusion type problems have been derived in [28,29]. For more complicated plate models in the theory of linear elasticity, such type estimates have been recently derived in [24].…”