  This paper focuses on the design, analysis and implementation of a new
preconditioning concept for linear second order partial differential equations,
including the convection-diffusion-reaction problems discretized by Galerkin or
discontinuous Galerkin methods. We expand on the approach introduced by
Gergelits et al. and adapt it to the more general settings, assuming that both
the original and preconditioning matrices are composed of sparse matrices of
very low ranks, representing local contributions to the global matrices. When
applied to a symmetric problem, the method provides bounds to all individual
eigenvalues of the preconditioned matrix. We show that this preconditioning
strategy works not only for Galerkin discretization, but also for the
discontinuous Galerkin discretization, where local contributions are associated
with individual edges of the triangulation. In the case of non-symmetric
problems, the method yields guaranteed bounds to real and imaginary parts of
the resulting eigenvalues. We include some numerical experiments illustrating
the method and its implementation, showcasing its effectiveness for the two
variants of discretized (convection-)diffusion-reaction problems.

Numerical Analysis;Numerical Analysis