We present a new algebraic multigrid (AMG) algorithm for the solution of linear systems arising from discontinuous Galerkin (DG) discretizations of heterogeneous elliptic problems. The algorithm is based on the idea of subspace corrections, and the first coarse level space is the subspace spanned by continuous linear basis functions. The linear system associated with this space is constructed algebraically using a Galerkin approach with the natural embedding as the prolongation operator. This embedding operator needs to be provided, which means that the approach is not fully algebraic. For the construction of the linear systems on the subsequent coarser levels, non-smoothed aggregation AMG techniques are used. In a series of numerical experiments, we establish for the first time the efficiency and robustness of an AMG method for various symmetric and non-symmetric interior penalty DG methods (including the higher-order cases) on problems with complicated, high contrast jumps in the coefficients. The solver is robust with respect to an increase in the polynomial degree of the DG approximation space (at least up to degree 6), computationally efficient, and affected only mildly by the coefficient jumps and by the mesh size h (i.e., number?of?iterations?=?O(log h-1)).
Bastian, P., Blatt, M., & Scheichl, R. (2012). Algebraic multigrid for discontinuous Galerkin discretizations of heterogeneous elliptic problems. Numerical Linear Algebra with Applications, 19(2), 367-388. https://doi.org/10.1002/nla.1816