Compatibility conditions for finite element approximations

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In the classical theory of the finite element approximation of elliptic partial differential equations, based on standard Galerkin schemes, the energy norm of the error decays with the same rate of convergence as the best finite element approximation, without any additional requirements on the involved spaces.
A different situation occurs when so called mixed finite element approximations are considered, for instance, for the approximation of Stokes or Darcy's flows. In such cases the finite element spaces have to satisfy suitable compatibility conditions: the famous inf-sup conditions. We review these conditions with the help of some one-dimensional examples.

Brief Biography

Daniele Boffi joined KAUST almost two years ago. Before joining KAUST he was professor of numerical analysis at the University of Pavia in Italy. His research is devoted to the numerical approximation of partial differential equations, with particular interest in the finite element method and in mixed finite elements.
Some of his most active research areas concern the approximation of eigenvalue problems arising from partial differential equations and the numerical modeling of fluid-structure interaction problems. He is coauthor of one of the most widely used book on mixed finite elements and he is author of a highly cited survey on the approximation of eigenvalue problems.

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