Equilibrated Averaging Residual Method: A General Approach to Conservative Flux Recovery

Many equilibrated flux recovery methods for finite element solutions rely on ad hoc or method-specific techniques, limiting their generalizability and efficiency. In this work, we introduce the Equilibrated Averaging Residual Method (EARM), a unified framework for flux recovery that not only reproduces state-of-the-art locally conservative fluxes but also enables the derivation of new equilibrated fluxes with improved properties. In this paper, EARM is applied to conforming, nonconforming, and discontinuous Galerkin methods, ensuring local conservation and robust a posteriori error estimation. Despite the unified nature of the variational problem, the framework retains the flexibility to fully leverage the inherent properties of finite element spaces. Moreover, EARM offers explicit and computationally efficient flux reconstructions for all methods in two dimensions. In three dimensions, only simple local problems need to be solved for the conforming finite element methods.