# Observations on degenerate saddle point problems

## TL;DR

This paper analyzes degenerate saddle point problems, proving their well-posedness under generalized conditions even with coefficient jumps, and connects these conditions to subspace gap measures, aiding numerical simulations of composite materials.

## Contribution

It introduces generalized conditions for well-posedness of degenerate saddle point problems, extending classical stability criteria to cases with large coefficient jumps.

## Key findings

- Proves well-posedness of degenerate saddle point problems in standard norm.
- Revisits and generalizes Ladygenskaya--Babuška--Brezzi conditions.
- Connects coercivity to the minimum gap of relevant subspaces.

## Abstract

We investigate degenerate saddle point problems, which can be viewed as limit cases of standard mixed formulations of symmetric problems with large jumps in coefficients. We prove that they are well-posed in a standard norm despite the degeneracy. By wellposedness we mean a stable dependence of the solution on the right-hand side. A known approach of splitting the saddle point problem into separate equations for the primary unknown and for the Lagrange multiplier is used. We revisit the traditional Ladygenskaya--Babu\v{s}ka--Brezzi (LBB) or inf--sup condition as well as the standard coercivity condition, and analyze how they are affected by the degeneracy of the corresponding bilinear forms. We suggest and discuss generalized conditions that cover the degenerate case. The LBB or inf--sup condition is necessary and sufficient for wellposedness of the problem with respect to the Lagrange multiplier under some assumptions. The generalized coercivity condition is necessary and sufficient for wellposedness of the problem with respect to the primary unknown under some other assumptions. We connect the generalized coercivity condition to the positiveness of the minimum gap of relevant subspaces, and propose several equivalent expressions for the minimum gap. Our results provide a foundation for research on uniform wellposedness of mixed formulations of symmetric problems with large jumps in coefficients in a standard norm, independent of the jumps. Such problems appear, e.g., in numerical simulations of composite materials made of components with contrasting properties.