An Algebraic Preconditioner for the Exactly Divergence-Free Discontinuous Galerkin Method for Stokes

We present an algebraic preconditioner for the exactly divergence-free discontinuous Galerkin (DG) discretization of Cockburn, Kanschat, and Sch & ouml;tzau [J. Sci. Comput., 31 (2007), pp. 61-73] and Wang and Ye [SIAM J. Numer. Anal., 45 (2007), pp. 1269-1286] for the Stokes problem. The exactly divergence-free DG method uses finite elements that use an H(div)$$ H\left(\operatorname{div}\right) $$-conforming basis, thereby significantly complicating its solution by iterative methods. Several preconditioners for this Stokes discretization has been developed, but each is based on specialized solvers or decompositions. To avoid requiring custom solvers, we hybridize the H(div)$$ H\left(\operatorname{div}\right) $$-conforming finite element so that the velocity lives in a standard L2$$ {L}<^>2 $$-DG space, and present a simple algebraic preconditioner for the extended hybridized system. The proposed preconditioner is optimal in mesh size h$$ h $$, effective in 2d and 3d, and only relies on standard relaxation and algebraic multigrid methods available in many packages. Furthermore, the Schur complement approximation is robust in element order k$$ k $$, although more AMG cycles are needed on the velocity block when increasing k$$ k $$.