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

## TL;DR

This paper introduces a systematic approach to designing bilinear forms that stabilize saddle point problems by ensuring spectral equivalence to scalar products, facilitating decoupled discretization strategies.

## Contribution

It provides a novel method to construct computable bilinear forms that stabilize saddle point problems and relax the inf-sup compatibility constraints.

## Key findings

- Bilinear forms are spectrally equivalent to scalar products on dual subspaces.
- Stabilized discretization algorithms decouple approximation and compatibility requirements.
- The approach enhances the flexibility of discretization choices in saddle point problems.

## Abstract

We provide a systematic way to design computable bilinear forms which, on the class of subspaces $W^* \subseteq \mathcal{V}'$ 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 $\mathcal{V}'$. Such a bilinear form 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 inf-sup compatibility condition, which, as we show, can not be completely avoided.