A Robust and Flexible A Posteriori Error Estimate for Elliptic Problems in $\mathbb{R}3$

Joint work with Michael Holst and Ryan Szypowski (both from the University of California, San Diego)

We propose and offer effectivity analysis of an a posteriori error estimate for tetrahedral linear Lagrange finite elements. The error estimate is based on the (provably inexpensive) computation of an approximate error function in an auxiliary space. We will provide an equivalence (up to oscillation terms) theorem of the true and approximate $H1$-error, which applies for piecewise smooth coefficients. In particular, our analysis applies to cases in which convection is present (non-coercive problems are fine) which distinguishes it from previous attempts in this direction. Numerical experiments demonstrate the robustness of the estimator with respect to: low regularity of the solution, discontinuities or anisotropies in the coefficients of the differential operator, and dominant convection. The nature of analysis suggests a broader applicability of the general error estimation approach---for example, to higher-order Lagrange elements or other types of elements. Finally, we will discuss a variety of applications of the approximate error function and of the general methodology, such as: functional error estimation, error estimation in other norms, error estimation and convergence of acceleration for eigenvalue problems, and selection of auxiliary spaces in which to compute approximate error functions for different finite elements. Some aspects have been proven, and others are work in progress. The numerical experiments presented in this part of the talk will be for problems in $\mathbb{R}2$.