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Preconditioning methods for local discontinuous Galerkin discretizations. (English) Zbl 1048.65110

A general framework for preconditioning saddle point problems has been successfully applied to stable, conforming discretizations of Stokes and Oseen equations by H. C. Elman, D. J. Silvester and A. J. Wathen [Numer. Math. 90, 665–688 (2002; Zbl 1143.76531)]. The authors use this approach to extend the multilevel preconditioner given by J. Gopalakrishnan and G. Kanschat [Numer. Math. 95, 527–550 (2003; Zbl 1044.65084)] to the saddle point matrices arising from the local discontinuous Galerkin discretization of Poisson equations and Stokes equations. In the Poisson case, a sharp estimate on the spectrum of the preconditioned matrix is obtained. Computational results are also obtained using the deal.II library for finite element computations.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
65F35 Numerical computation of matrix norms, conditioning, scaling
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35Q30 Navier-Stokes equations

Software:

deal.ii
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