The Prager–Synge theorem in reconstruction based a posteriori error estimation

Abstract

In this paper we review the hypercircle method of Prager and Synge. This theory inspired several studies and induced an active research in the area of a posteriori error analysis. In particular, we review the Braess-Schöberl error estimator in the context of the Poisson problem. We discuss adaptive finite element schemes based on two variants of the estimator and we prove the convergence and optimality of the resulting algorithms.

Guaranteed upper bounds for the velocity error of pressure-robust Stokes discretisations

“…A Prager-Synge-type result for the Stokes system. This section states a Pythagoras theorem for the Stokes system similar to that of Prager and Synge for the Poisson model problem and the linear elasticity problem [35,2]. The Prager-Synge theorem relates the error of primal and equilibrated mixed approximations of the flux ∇u (or ǫpuq in elasticity) and gives rise to guaranteed error control by the design of equilibrated fluxes for these problems.…”

“…However, care has to be taken for the additional divergence constraint that often leads to pressure-dependent velocity error estimators or estimators for the combined velocity and pressure error. For problems of the form (1), there is the famous Prager-Synge theorem [35,2] (originally for linear elasticity) that is nothing else than a Pythagoras theorem in L 2 -norms, i.e. }∇pu ´vq} 2 `}∇u ´ν´1 σ} 2 " }∇v ´ν´1 σ} 2 , where u can be understood as some approximation to u and σ only has to satisfy some orthogonality or equilibration constraint.…”

“…For problems of the form (1), there is the famous Prager-Synge theorem [35,2] (originally for linear elasticity) that is nothing else than a Pythagoras theorem in L 2 -norms, i.e. }∇pu ´vq} 2 `}∇u ´ν´1 σ} 2 " }∇v ´ν´1 σ} 2 , where u can be understood as some approximation to u and σ only has to satisfy some orthogonality or equilibration constraint. In our Stokes setting it is required that u, v P V 0 and ż Ω pdiv σ `f q ¨w dx " 0 for all w P V 0 (2) where V 0 is the subspace of divergence-free H 1 0 pΩq test functions.…”

“…}∇pu ´vq} 2 `}∇u ´ν´1 σ} 2 " }∇v ´ν´1 σ} 2 , where u can be understood as some approximation to u and σ only has to satisfy some orthogonality or equilibration constraint. In our Stokes setting it is required that u, v P V 0 and ż Ω pdiv σ `f q ¨w dx " 0 for all w P V 0 (2) where V 0 is the subspace of divergence-free H 1 0 pΩq test functions. An important observation is that, opposite to the Poisson problem or linear elasticity where the constraint has to hold for the whole space H 1 0 pΩq, equation ( 2) is not equivalent to divpσq `f " 0.…”

“…To remove this dependency we propose a novel equilibration design that avoids the gauging issue altogether and ensures the equilibration condition (2) as it is for an Hpdivq-conforming subspace of V 0 . We so ensure that even after the discretisation of the equilibration constraint, the complete gauging freedom is preserved.…”