Exact Bounds for Linear Outputs of the Convection-Diffusion-Reaction Equation Using Flux-Free Error Estimates

Abstract

The Flux-free approach is a promising alternative to standard implicit residual error estimators that require the equilibration of hybrid fluxes. The idea is to solve local error problems in patches of elements surrounding one node (stars) instead of in single elements [1]. The resulting local problems are flux-free, that is the boundary conditions are natural and hence their implementation is straightforward. This allows precluding the computation and the equilibration of fluxes along the element edges. The domain decomposition is performed using a partition of unity strategy. The resulting estimates are much simpler from the implementation viewpoint, especially in the 3D cases, and provide upper bounds of the energy norm of the error (as well as the standard implicit residual estimators with equilibration of hybrid fluxes).

In the past, the local flux-free problems have been solved using a finite element mesh inside each local subdomain. Consequently, the resulting estimates were asymptotic upper bounds (w.r.t a reference solution) rather than exact upper bounds (w.r.t the exact solution). Some effort has been devoted to recover exact upper bounds using the equilibrated hybrid fluxes approach. The idea is to solve the local problem using a dual formulation and to minimize the complementary energy [2]. In this work, the same idea is employed to obtain exact upper bounds using the flux-free approach. The resulting estimates have similar features as their asymptotic version, while providing a guaranteed upper bound. This strategy is applied both to the primal and adjoint problem to recover guaranteed bounds for the quantity of interest.