A posteriori error analysis for locally conservative mixed methods

Abstract

In this work we present a theoretical analysis for a residual-type error estimator for locally conservative mixed methods. This estimator was first introduced by Braess and Verfürth for the Raviart–Thomas mixed finite element method working in mesh-dependent norms. We improve and extend their results to cover any locally conservative mixed method under minimal assumptions, in particular, avoiding the saturation assumption made by Braess and Verfürth. Our analysis also takes into account discontinuous coefficients with possibly large jumps across interelement boundaries. The main results are applied to the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Baseline 1"> <mml:semantics> <mml:mrow> <mml:mi>P</mml:mi> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">P1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> nonconforming finite element method and the interior penalty discontinuous Galerkin method as well as the mixed finite element method.