Preconditioning Methods for Local Discontinuous Galerkin Discretizations

A multilevel interior penalty method is used as an efficient preconditioner for the Schur complement of the local discontinuous Galerkin (LDG) discretization of a Poisson problem. The method is then used in a block-triangular preconditioner of the LDG saddle point system. The block preconditioner is of the same efficiency as the Schur complement version. Finally, the block preconditioner is extended to the discretization of the Stokes problem by the LDG method. Again, the preconditioned saddle point problem can be solved in about as many steps as the Schur complement. The influence of several parameters on the performance of these methods is investigated.