Computational aspects of the Local Discontinuous Galerkin method: an algorithmic approach

The Local Discontinuous Galerkin (LDG) method is one of several discontinuous Galerkin (DG) methods that has been extensively studied in recent years. In this presentation we discuss several computational issues. After a brief introduction of the method applied to a second order linear operator, we describe the general structure of the discrete linear system and discuss the influence of the method’s parameters. In particular we address some issues about the stencil of the global stiffness matrix which has been pointed out in the literature as the major drawback compared to other DG methods which have a compact stencil. It has been recently proven that the spectral condition number of the stiffness matrix exhibits an asymptotic behavior of O(h) on structured and unstructured meshes, where h is the mesh size. Thus efficient preconditioners are mandatory. We present a semi-algebraic multilevel preconditioner for the LDG method using local Lagrange type interpolatory basis functions. We show, numerically, that its performance does not degrade, or at least the number of iterations increases very slowly, as the number of unknowns augments. The preconditioner is tested on problems with high jumps in the coefficients, which is the typical scenario of problems arising in porous media. Finally we present an a posteriori global error estimate for the LDG method applied to a linear second order elliptic problem. Using a mixed formulation, an upper bound of the error in the primal variable is derived from explicit computations. A local adaptive scheme based on explicit error estimators is studied numerically. Department of Mathematical Sciences 31 January 2008