The Local Discontinuous Galerkin (LDG) method is one of several Discontinuous Galerkin (DG) methods which has been extensively studied in recent years. Several papers have been devoted to the analysis of the stability and convergence of the method applied to linear and nonlinear diffusion problems and high order differential operators. In this talk we discuss several
aspects of the LDG method applied to linear second order elliptic problems on unstructured meshes in 3D.

We briefly describe the method using a general abstract framework in which several DG methods can be formulated. A simplified description of the most relevant operators is formulated using tensor notation. A reduction of storage is achieved by introducing a fast algorithm for the assembly of the Schur complement explicitely. The flexibility of the code relies on the design of abstract data structures. A series of numerical experiments are presented to validate the code and to illustrate the performance of the method on unstructured meshes in 3D.

It has been proven that the spectral condition number of the stiffness matrix exhibits an asymptotic behavior of O(h-2) on structured and unstructured meshes where h is the mesh size. Thus efficient preconditioners are mandatory. We present semi-algebraic multilevel preconditioners for linear approximations that use Lagrange type interpolatory basis. We show numerically that their performance does not degrade or at least increases very slowly as the number of unknowns augment. Preconditioners are tested on problems with high jumps in the coefficients, which is the typical scenario of problems arising in porous media.

Classification: 65N30, 65N12, 65N55, 65F08

Keywords Discontinuous Galerkin methods, high order approximations, multilevel preconditioners.