Adaptive virtual element methods with equilibrated fluxes

Adaptive Finite Element Methods

In the numerical solution of practical problems of physics or engineering such as, e.g., computational fluid dynamics, elasticity, or semiconductor device simulation one often encounters the difficulty that the overall accuracy of the numerical approximation is deteriorated by local singularities such as, e.g., singularities arising from re-entrant corners , interior or boundary layers, or shar...

Adaptive Finite Element Methods

Adaptive methods are now widely used in the scientific computation to achieve better accuracy with minimum degree of freedom. In this chapter, we shall briefly survey recent progress on the convergence analysis of adaptive finite element methods (AFEMs) for second order elliptic partial differential equations and refer to Nochetto, Siebert and Veeser [14] for a detailed introduction to the theo...

Adaptive Finite Element Methods with convergence rates

Algebraic and Discretization Error Estimation by Equilibrated Fluxes for Discontinuous Galerkin Methods on Nonmatching Grids

We derive a posteriori error estimates for the discontinuous Galerkin method applied to the Poisson equation. We allow for a variable polynomial degree and simplicial meshes with hanging nodes and propose an approach allowing for simple (nonconforming) flux reconstructions in such a setting. We take into account the algebraic error stemming from the inexact solution of the associated linear sys...

A Posteriori Error Estimators Based on Equilibrated Fluxes

We consider the conforming of finite element approximations of reactiondiffusion problems. We propose new a posteriori error estimators based on H(div)conforming finite elements and equilibrated fluxes. It is shown that these estimators give rise to an upper bound where the constant is one in front of the indicator, up to higher order terms. Lower bounds can also be established with constants d...