Robust BDDC algorithms for the Brinkman problem with HDG discretizations

Abstract

The balancing domain decomposition by constraints methods (BDDC) have been applied to solve the saddle point problem arising from a hybridizable discontinuous Galerkin (HDG) discretization for the Brinkman equations. Edge/face average constraints are enforced across the subdomain interface for each velocity component to ensure that the BDDC preconditioned conjugate gradient (CG) iterations stay in a special subspace, where the reduced system from the original saddle point problem is positive definite. The condition number is proved to be uniformly bounded under both Stokes and Darcy dominant cases with a deluxe scaling for the parameters with jumps only across the subdomain interface. When the parameters are highly discontinuous with large jumps inside each subdomains, additional adaptively chosen primal constraints, obtained by solving local generalized eigenvalue problems, are introduced to control the condition numbers. Numerical experiments confirm the theory.