An element-based preconditioner for mixed finite element problems

We introduce a new and generic approximation to Schur complements arising from inf-sup stable mixed finite element discretizations of self-adjoint multi-physics problems. The approximation exploits the discretization mesh by forming local, or element, Schur complements and projecting them back to the global degrees of freedom. The resulting Schur complement approximation is sparse, has low construction cost (with the same order of operations as a general finite element matrix), and can be solved using off-the-shelf techniques, such as multigrid. Using the Ladyshenskaja-Babu\v{s}ka-Brezzi condition, we show that this approximation has favorable eigenvalue distributions with respect to the global Schur complement. We present several numerical results to demonstrate the viability of this approach on a range of applications. Interestingly, numerical results show that the method gives an effective approximation to the non-symmetric Schur complement from the steady state Navier-Stokes equations.