A posteriori Error Estimates with Post-Processingfor Nonconforming Finite Elements

Abstract

For a nonconforming finite element approximation of an elliptic model problem, we propose a posteriori error estimates in the energy norm which use as an additive term the “post-processing error” between the original nonconforming finite element solution and an easy computable conforming approximation of that solution.Thus, for the error analysis, the existing theory from the conformingcase can be used together with some simple additional arguments.As an essential point, the property is exploited that the nonconformingfinite element space contains as a subspace a conforming finite element space of first order. This property is fulfilled for many knownnonconforming spaces. We prove local lower and global upper a posteriori error estimates for an enhanced error measure which is the discretization error in the discrete energy norm plus the error of the best representation of the exact solution by a function in the conforming space used for thepost-processing. We demonstrate that the idea to use a computed conforming approximation ofthe nonconforming solution can be applied also to derive an a posteriorierror estimate for a linear functional of the solution which representssome quantity of physical interest.