Algebraic representation of dual scalar products and stabilization of saddle point problems

Abstract

We provide a systematic way to design computable bilinear forms which, on the class of subspaces $W\subseteq {\mathcal{V}}^{\prime }$ that can be obtained by duality from a given finite dimensional subspace $W$ of an Hilbert space $\mathcal{V}$, are spectrally equivalent to the scalar product of ${\mathcal{V}}^{\prime }$. In the spirit of Baiocchi–Brezzi (1993) and Bertoluzza (1998), such bilinear forms can be used to build a stabilized discretization algorithm for the solution of an abstract saddle point problem allowing to decouple, in the choice of the discretization spaces, the requirements related to the approximation from the ones related to the inf-sup compatibility condition, which, however, can not be completely avoided.