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

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--Sch\"oberl 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

The DPG-star method

This article introduces the DPG-star (from now on, denoted DPG$^*$) finite element method. It is a method that is in some sense dual to the discontinuous Petrov-Galerkin (DPG) method. The DPG methodology can be viewed as a means to solve an overdetermined discretization of a boundary value problem. In the same vein, the DPG$^*$ methodology is a means to solve an underdetermined discretization. These two viewpoints are developed by embedding the same operator equation into two different saddle-point problems. The analyses of the two problems have many common elements. Comparison to other methods in the literature round out the newly garnered perspective. Notably, DPG$^*$ and DPG methods can be seen as generalizations of $\mathcal{L}\mathcal{L}^\ast$ and least-squares methods, respectively. A priori error analysis and a posteriori error control for the DPG$^*$ method are considered in detail. Reports of several numerical experiments are provided which demonstrate the essential features of the new method. A notable difference between the results from the DPG$^*$ and DPG analyses is that the convergence rates of the former are limited by the regularity of an extraneous Lagrange multiplier variable

Two-sided a posteriori error estimates for mixed formulations of elliptic problems

The present work is devoted to the a posteriori error estimation for mixed approximations of linear self-adjoint elliptic problems. New guaranteed upper and lower bounds for the error measured in the natural product norm are derived, and individual sharp upper bounds are obtained for approximation errors in each of the physical variables. All estimates are reliable and valid for any approximate solution from the class of admissible functions. The estimates contain only global constants depending solely on the domain geometry and the given operators. Moreover, it is shown that, after an appropriate scaling of the coordinates and the equation, the ratio of the upper and lower bounds for the error in the product norm never exceeds 3. The possible methods of finding the approximate mixed solution in the class of admissible functions are discussed. The estimates are computationally very cheap and can also be used for the indication of the local error distribution. As applications, the diffusion problem as well as the problem of linear elasticity are considered