## 00-11

#### A Posteriori Error Estimates with Post-Processing for Nonconforming Finite Elements

**Preprint series:** 00-11, Preprints

**MSC****:**- 65N15 Error bounds
- 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods

**Abstract:** For a nonconforming finite element approximation of an elliptic model problem, we propose an a posteriori error estimate in the energy norm which uses 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 for the conforming case can be used together with some simple additional arguments. As an essential point, the property is exploited that the nonconforming finite element space contains as a subspace a conforming finite element space of first order. This property is fulfilled for many known nonconforming spaces. For the a posteriori error bound, we prove that it has the same asymptotic behavior as the energy norm of the real discretization error itself. We show that the \'\'post-processing error\'\' can be used also as an additional error indicator. Besides the error estimates in the global energy norm, we demonstrate that the concept of using a conforming approximation of the nonconforming solution can be applied also to derive an a posteriori error estimate for linear functionals of the solution which represent some quantities of physical interest.

**Keywords:** a posteriori error estimates, nonconforming finite elements, post-processing

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