Adaptive Coarse Space for Saddle Point Problem

  • Nataf , Frédéric (Laboratoire J.L. Lions)
  • Tournier, Pierre-Henri (Laboratoire J.L. Lions)

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We introduce a scalable adaptive element-based domain decomposition (DD) method for solving saddle point problems defined as a block two by two matrix. Such linear systems are ubiquitous in scientific computing. The stable discretization of equations for (nearly) incompressible fluids or elastic bodies requires a velocity-pressure formulation. Also in many codes Multi Point Constraints (MPC) are enforced with Lagrange multipliers. A 2x2 block system structure represents as well many characteristics of multiphysics systems. The upper left matrix contains the various physics and the off-diagonal part the coupling conditions between the physics. Our algorithm does not require any knowledge of the constrained space. We assume that all sub matrices are sparse and that the diagonal blocks are spectrally equivalent to a sum of positive semi definite matrices. The latter assumption enables the design of adaptive coarse space for DD methods that extends the GenEO theory to saddle point problems. Numerical results on three dimensional elasticity problems for steel-rubber structures discretized by a finite element with continuous pressure are shown for up to one billion degrees of freedom along with comparisons to Algebraic Multigrid Methods and direct solvers. The method has also been applied to Stokes system with discontinuous coefficients.