Numerical and computational aspects of some block-preconditioners for saddle point systems

• Element-wise approximation of the Schur complement (EWS) for saddle point matrices.
• Spectral bounds are shown when the Schur complement is symmetric negative definite.
• EWS leads to a numerically and computationally optimal iterative solver.
• Numerical simulations on various computer architectures.

Linear systems with two-by-two block matrices are usually preconditioned by block lower- or upper-triangular systems that require an approximation of the related Schur complement. In this work, in the finite element framework, we consider one special such approximation, namely, the element-wise Schur complement. It is sparse and its construction is perfectly parallelizable, making it an appropriate ingredient when building preconditioners for iterative solvers executed on both distributed and shared memory computer architectures. For saddle point matrices with symmetric positive (semi-)definite blocks we show that the Schur complement is spectrally equivalent to the so-constructed approximation and derive spectral equivalence bounds. We also illustrate the quality of the approximation for nonsymmetric problems, where we observe the same good numerical efficiency.Furthermore, we demonstrate the computational and numerical performance of the corresponding preconditioned iterative solution method on a large scale model benchmark problem originating from the elastic glacial isostatic adjustment model discretized using the finite element method.