| We present a Saddle Point Least Squares (SPLS) method for solving variational formulations with different types of trial and test spaces. The general mixed formulation we consider assumes a stability LBB condition and a data compatibility condition at the continuous level. We expand on the Bramble-Pasciak's least square formulation for solving such problems by providing new ways to choose approximation spaces and new iterative processes to solve the discrete formulations. The proposed discretization method follows a general SPLS approach and has the advantage that a discrete inf -- sup condition is automatically satisfied for the standard choices of the test and trial spaces. For the proposed iterative processes a nodal basis for the trial space is not required and efficient preconditioning techniques that involve inversion only on the test space can be considered. Stability and approximation properties for two choices of discrete spaces are investigated. Applications of the new approach include discretization of first order systems of PDEs, such as div -- curl systems, second order problems with highly oscillatory coefficient, interface problems, and higher order approximation of the flux for elliptic problems with smooth coefficients. |
