A Weakly Penalized Discontinuous Galerkin
Method for Radiation in Dense, Scattering
Media
We review the derivation of weakly penalized
discontinuous Galerkin methods for scattering
dominated radiation transport and extend the
asymptotic analysis to non-isotropic scattering.
Multilevel Schwarz preconditioners for singularly
perturbed symmetric reaction-diffusion systems
We present robust and highly parallel multilevel
non-overlapping Schwarz preconditioners, to solve an
interior penalty discontinuous Galerkin finite element
discretization of a system of steady state, singularly
perturbed reaction-diffusion equations with a singular
reaction operator, using a GMRES solver. We provide proofs
of convergence for the two-level setting and the multigrid
V-cycle as well as numerical results for a wide range of
regimes.
Should multilevel methods for discontinuous Galerkin
discretizations use discontinuous interpolation operators?
We consider a discontinuous interpolation operator
with a parameter that controls the discontinuity, and determine the
optimal choice for the discontinuity, leading to the fastest solver
for a specific 1D symmetric interior penalty DG discretization model
problem. We show in addition that our optimization delivers a
perfectly clustered spectrum with a high geometric multiplicity, which
is very advantageous for a Krylov solver using the method as its
preconditioner.
Optimization of two-level methods for DG discretizations of
reaction-diffusion equations
We apply two-level methods to a symmetric interior penalty
discontinuous Galerkin finite element discretization of a
singularly perturbed reactions-diffusion equation.
The main innovation is that explicit formulas for the
optimal relaxation parameter of the two-level method for the
Poisson problem in 1D are obtained, as well as very accurate
closed form approximation formulas for the optimal choice in
the reaction-diffusion case in all regimes.
Numerical experiments and comparisons show the applicability
of the expressions obtained in higher dimensions and general
geometries.