Black-Box Preconditioning for Mixed Formulation of Self-Adjoint Elliptic PDEs
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.
KeywordsMixed Finite Element Mixed Finite Element Method Diagonal Scaling Precondition Strategy Mixed Finite Element Approximation
Unable to display preview. Download preview PDF.
- Bramble, J.H.: Multigrid Methods. Pitman Research Notes in Mathematics Series, 294, Longman (1993).Google Scholar
- Powell, C.E., Silvester, D.J.: Optimal preconditioning for Raviart-Thomas mixed formulation of second-order elliptic problems. Manchester Centre for Computational Mathematics Report, 399, (2002).Google Scholar
- Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations, Springer-Verlag, (1994).Google Scholar
- Raviart, P.A., Thomas, J.A.: A mixed finite element method for second order elliptic problems. Mathematical Aspects of the Finite Element Method. Lecture Notes in Mathematics, 606. Springer-Verlag, New York (1977).Google Scholar
- Ruge, J.W., Stäben, K.: Efficient solution of finite difference and finite element equations by algebraic multigrid (AMG). In: Paddon, D.J., Holstein, H., (eds.): Multigrid Methods for Integral and Differential Equations. The Institute of Mathematics and its Applications Conference Series. Oxford, (1985) 169–212Google Scholar
- Silvester, D., Wathen, A.: Fast and robust solvers for time-discretised incompressible Navier-Stokes equations. In: Griffiths, D.F., Watson, G.A, (eds.): Numerical Analysis 1995. Pitman Research Notes in Mathematics Series, 344, Longman (1996).Google Scholar