An object-oriented framework for hp-adaptive discontinuous galerkin methods

Abstract

In this work, we consider second order elliptic problems arising in the modeling of single phase flows in porous media in 2D and in the analysis of transverse electromagnetic modes in wave guides using a discontinuous Galerkin (DG) method, the so-called Local Discontinuous Galerkin (LDG) method. We designed and developed an object oriented framework for performing DG computations on unstructured meshes that allows the use of arbitrary degree in the polynomial approximations and non conformal meshes with an arbitrary number of hanging nodes per edge. We present numerical studies of an automatic mesh adaptation technique and a semi-algebraic multilevel preconditioner for the LDG method. DG methods may be viewed as high-order extensions of the classical finite volume method. Since no inter-element continuity is imposed, they can be defined on very general meshes, including non-conforming meshes, making these methods suitable for h-adaptivity. Our adaptive algorithm starts with an initial, conformal spatial discretization of the domain where the numerical solution of the partial differential equation is obtained using the LDG method. In each step, the error of the solution is estimated and the mesh is modified successively by performing two local operations: refining a fraction of the cells where the estimated error is greater and agglomerating a fraction of the cells where the estimated error is smallest. This procedure is repeated until the solution reaches a desired accuracy. It has been recently shown 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, making the use of effective preconditioners a practical requirement. We present a semi-algebraic multilevel preconditioner for the LDG method and show through several numerical experiments that its performance does not degrade as the number of unknowns augments. The performance of these techniques is explored on problems with high jumps in the coefficients, which is the typical scenario of problems arising in practical applications.