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This paper focuses on the design, analysis and implementation of a newpreconditioning concept for linear second order partial differential equations,including the convection-diffusion-reaction problems discretized by Galerkin ordiscontinuous Galerkin methods. We expand on the approach introduced byGergelits et al. and adapt it to the more general settings, assuming that boththe original and preconditioning matrices are composed of sparse matrices ofvery low ranks, representing local contributions to the global matrices. Whenapplied to a symmetric problem, the method provides bounds to all individualeigenvalues of the preconditioned matrix. We show that this preconditioningstrategy works not only for Galerkin discretization, but also for thediscontinuous Galerkin discretization, where local contributions are associatedwith individual edges of the triangulation. In the case of non-symmetricproblems, the method yields guaranteed bounds to real and imaginary parts ofthe resulting eigenvalues. We include some numerical experiments illustratingthe method and its implementation, showcasing its effectiveness for the twovariants of discretized (convection-)diffusion-reaction problems.

Preconditioned discontinuous Galerkin method and convection-diffusion-reaction problems with guaranteed bounds to resulting spectra