Enhanced functional evaluation for the finite element penalty method

Abstract

We analyze the impact of using the penalty method on the estimation of a prescribed solution functional or ‘Quantity of Interest’ (QoI). Specifically, we consider the use of penalty methods to enforce Dirichlet boundary constraints, focusing our attention on boundary fluxes as QoIs. We propose an enhanced estimator of the boundary flux that includes a term involving the derivative of the flux with respect to the penalty parameter ϵ. We show that the new estimator reduces the error arising from the use of the penalty method from O(ϵ) to o(ϵ). A well-posed adjoint problem associated with the boundary flux is also proposed following an analysis of the penalty method. Errors in the enhanced flux estimator are then controlled using adjoint-based techniques. Several numerical experiments are presented to demonstrate that the enhanced estimator, in combination with adjoint error estimation, allows one to efficiently control both the discretization and penalty errors in target QoIs.