A Posteriori Estimation of Dimension Reduction Errors for Elliptic Problems on Thin Domains

Abstract

A new a posteriori error estimator is presented for the verification of the dimensionally reduced models stemming from the elliptic problems on thin domains. The original problem is considered in a general setting, without any specific assumptions on the domain geometry, coefficients, and the right-hand sides. For the energy norm of the error of the zero-order dimension reduction method, the proposed estimator is shown to always provide a guaranteed upper bound. In the case when the original domain has constant thickness (but, possibly, nonplane upper and lower faces), the estimator demonstrates the optimal convergence rate as the thickness tends to zero. It is also flexible enough to successfully cope with infinitely growing right-hand sides in the equation when the domain thickness tends to zero. The numerical tests indicate the efficiency of the estimator and its ability to accurately represent the local error distribution needed for an adaptive improvement of the reduced model.