Elliptic Problems with Adaptivity

Abstract

In this chapter we investigate and apply adaptive dG algorithms for the stationary semi-linear ADR equations of the model (1.1). We give the existence and uniqueness results of the elliptic system. The main focus of this chapter is handling of unphysical oscillations at the interior/boundary layers in advection dominated problems resulting through the discretization in space by applying an adaptive algorithm using residual-based robust a posteriori error estimates for the stationary model. The results obtained in this chapter for the stationary model will be a key ingredient in the follow-up chapter for the non-stationary models. Since the stiffness matrices obtained by dG methods become more dense and ill-conditioned with increasing order of dG polynomials, the resulting linear system of equations have to preconditioned. For this reason, we introduce in this chapter the matrix reordering and iterative partitioning technique in [86]. We give the details of the construction of the matrix reordering and partitioning technique, and demonstrate its efficiency numerically.