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A note on stability and optimal approximation estimates for symmetric saddle point systems

We establish sharp well-posedness and approximation estimates for variational saddle point systems at the continuous level. The main results of this note have been known to be true only in the finite dimensional case. Known spectral results from the discrete case are reformulated and proved using a functional analysis view, making the proofs in both cases, discrete and continuous, less technical than the known discrete approaches. We focus on analyzing the special case when the form $a(\cdot, \cdot)$ is bounded, symmetric, and coercive, and the mixed form $b(\cdot, \cdot)$ is bounded and satisfies a standard $\inf-\sup$ or LBB condition. We characterize the spectrum of the symmetric operators that describe the problem at the continuous level. For a particular choice of the inner product on the product space of $b(\cdot, \cdot)$, we prove that the spectrum of the operator representing the system at continuous level is $\left \{\frac{1-\sqrt{5}}{2}, 1, \frac{1+\sqrt{5}}{2} \right \}$. As consequences of the spectral description, we find the minimal length interval that contains the ratio between the norm of the data and the norm of the solution, and prove explicit approximation estimates that depend only on the continuity constant and the continuous and the discrete $\inf-\sup$ condition constants.