Sharp Stability and 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(·, ·) is bounded, symmetric, and coercive, and the mixed form b(·, ·) 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(·, ·), we prove that the spectrum of the operator representing the system at continuous level is { 1− √ 5 2 , 1, 1+ √ 5 2 } . 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. 1. Notation and standard properties The existing literature on stability and approximation estimates for symmetric Saddle Point (SP) systems is quite rich for both continuous and discrete levels. While at discrete level the estimates can be done using eigenvalue analysis of symmetric matrices and consequently are optimal, at the continuous level, the estimates are presented as inequalities depending on related constants and consequently are not optimal. In this note, we will establish optimal estimates at the continuous level, that can be viewed as generalizations of results at the discrete level. The new spectral estimates provide more insight into the behavior of the general symmetric SP problems. In addition, information about the continuous spectrum and the techniques used to characterize it can lead to efficient analysis of iterative methods for SP systems. Towards this end, we let V and Q be two Hilbert spaces with inner products given by symmetric bilinear forms a(·, ·) and (·, ·) respectively, with 2000 Mathematics Subject Classification. 74S05, 74B05, 65N22, 65N55.