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The role of a posteriori error estimators in the approximation of some Boundary Value Problems

We present a short overview on a posteriori estimators for the error arising in the Finite Element approximation of Boundary Value Problems. In particular, we show how these techniques may improve the classical Finite Element Method by providing both qualitative and quantitative information.

First, we consider an image segmentation problem and we introduce residual-type and recovery-based error estimators. Within this framework, we highlight the advantages of using the qualitative information of the estimators to construct a computational mesh adapted to the problem and with a limited number of Degrees of Freedom.

A second example focuses on the quantitative role of a posteriori error estimators. We consider a shape optimization approach to the inverse problem of Electrical Impedance Tomography and we show that a certified fully-computable error estimator provides useful information to derive a guaranteed stopping criterion thus making the optimization strategy automatic.


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