An adaptive superconvergent mixed finite element method for exactly symmetric linear elasticity based on local residual minimization

Abstract

We introduce an a posteriori error estimator for exactly symmetric mixed finite element discretizations of linear elasticity, based on recasting a Stenberg-type postprocessing scheme as a local residual minimization problem in a discrete dual norm. This construction yields, as a dual variable and at no additional computational cost, a Riesz representative of the associated local residual, from which we build an a posteriori error indicator. We establish a reliability estimate, with the dependence on the Lam\'e parameters tracked explicitly, and a local efficiency estimate, without auxiliary bubble functions, in the standard compressible regime, as well as an alternative reliability estimate, based on a robust stability property and an Oswald averaging operator, with a constant that remains bounded as $\lambda\to\infty$; the local efficiency estimate holds, with the same bounded behavior, uniformly in both regimes. Notably, a single indicator and a single comparison norm serve both regimes, in contrast with existing hypercircle-based estimators, which require a distinct construction for the incompressible limit. Numerical examples, including a benchmark with a known singular solution and one without an analytical solution, validate the theoretical findings.