  In this paper, we develop a new tensor-product based preconditioner for
discontinuous Galerkin methods with polynomial degrees higher than those
typically employed. This preconditioner uses an automatic, purely algebraic
method to approximate the exact block Jacobi preconditioner by Kronecker
products of several small, one-dimensional matrices. Traditional matrix-based
preconditioners require $\mathcal{O}(p^{2d})$ storage and $\mathcal{O}(p^{3d})$
computational work, where $p$ is the degree of basis polynomials used, and $d$
is the spatial dimension. Our SVD-based tensor-product preconditioner requires
$\mathcal{O}(p^{d+1})$ storage, $\mathcal{O}(p^{d+1})$ work in two spatial
dimensions, and $\mathcal{O}(p^{d+2})$ work in three spatial dimensions.
Combined with a matrix-free Newton-Krylov solver, these preconditioners allow
for the solution of DG systems in linear time in $p$ per degree of freedom in
2D, and reduce the computational complexity from $\mathcal{O}(p^9)$ to
$\mathcal{O}(p^5)$ in 3D. Numerical results are shown in 2D and 3D for the
advection and Euler equations, using polynomials of degree up to $p=15$. For
many test cases, the preconditioner results in similar iteration counts when
compared with the exact block Jacobi preconditioner, and performance is
significantly improved for high polynomial degrees $p$.
