A new 3D equilibrated residual method improving accuracy and efficiency of flux‐free error estimates

Abstract

Summary The paper presents a novel strategy providing fully computable upper bounds for the energy norm of the error in the context of three‐dimensional linear finite element approximations of the reaction‐diffusion equation. The upper bounds are guaranteed regardless the size of the finite element mesh and the given data, and all the constants involved are fully computable. The upper bound property holds if the shape of the domain is polyhedral and the Dirichlet boundary conditions are piecewise‐linear. The new approach is an extension of the flux‐free methodology introduced by Parés and Díez in the paper “A new equilibrated residual method improving accuracy and efficiency of flux‐free error estimates”, which introduces a guaranteed, low‐cost, and efficient flux‐free method substantially reducing the computational cost of obtaining guaranteed bounds using flux‐free methods while retaining the good quality of the bounds. Besides extending the 2D methodology, specific new modifications are introduced to further reduce the computational cost in the three‐dimensional setting. The presented methodology also provides a new strategy to obtain equilibrated boundary tractions, which improves the quality of standard techniques while having a similar computational cost.

A posteriori goal-oriented bounds for the Poisson problem using potential and equilibrated flux reconstructions: application to the hybridizable discontinuous Galerkin method

“…Note that max x∈e |x − x e | can be replaced by h K and the inequalities still hold. The proof of these results can be found in [17,35,36,4].…”

“…For non-polynomial data, it is not possible in general to find reconstructions satisfying (9), and therefore (10) cannot be used to compute guaranteed bounds for the output. Fortunately, we can employ the technique described in [14,23,24,3] to recover bounds for the energy from projected equilibrated flux reconstructions by means of introducing data oscillation errors [17,35,36,4]. Let T h be a collection of d-dimensional non-overlaping and non-degenerate simplices K that partition Ω, such that the intersection of a distinct pair of elements is either an empty set or their common node, edge or face (in three dimensions).…”