  We study the convergence of iterative linear solvers for discontinuous
Galerkin discretizations of systems of hyperbolic conservation laws with
polygonal mesh elements compared with that of traditional triangular elements.
We solve the semi-discrete system of equations by means of an implicit time
discretization method, using iterative solvers such as the block Jacobi method
and GMRES. We perform a von Neumann analysis to analytically study the
convergence of the block Jacobi method for the two-dimensional advection
equation on four classes of regular meshes: hexagonal, square,
equilateral-triangular, and right-triangular. We find that hexagonal and square
meshes give rise to smaller eigenvalues, and thus result in faster convergence
of Jacobi's method. We perform numerical experiments with variable velocity
fields, irregular, unstructured meshes, and the Euler equations of gas dynamics
to confirm and extend these results. We additionally study the effect of
polygonal meshes on the performance of block ILU(0) and Jacobi preconditioners
for the GMRES method.
