Black-Box Preconditioning for Mixed Formulation of Self-Adjoint Elliptic PDEs

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Mixed finite element approximation of self-adjoint elliptic PDEs leads to symmetric indefinite linear systems of equations. Preconditioning strategies commonly focus on reduced symmetric positive definite systems and require nested iteration. This deficiency is avoided if preconditioned MINRES is applied to the full indefinite system. We outline such a preconditioning strategy, the key building block for which is a fast solver for a scalar diffusion operator based on black-box algebraic multigrid. Numerical results are presented for the Stokes equations arising in incompressible flow modelling and a variable diffusion equation that arises in modelling potential flow. We prove that the eigenvalues of the preconditioned matrices are contained in intervals that are bounded independently of the discretisation parameter and the PDE coefficients.