High-Order dicontinuous methods for linear convection-diffusion problems in 2D and 3D

Abstract

This thesis analyzes the advantages and drawbacks of several high-order finite element formulations to solve 2D and 3D steady-state and transient linear convection-diffusion problems. Continuous and discontinuous element-wise polynomial formulations are considered. The efficiency of traditional Galekin approach (CG) is compared to its statically condensed version (HCG) and a hybridizable local discontinuous Galerkin method (HDG). Latter method is the one that motivates the whole study because of its stability and superconvergence properties together with a hybridizable formulation, that promises reducing computational costs. In order to assess accuracy and computational performance, the three aforementioned numerical methods have been implemented using Python programming language. The postprocessed solution provided by HDG is shown to converge at p + 2 rates in the L 2 -norm for quadrilaterals and hexahedral elements. Moreover its stability capabilities are assessed for convection-dominant problems. Finally, hybridization and static condensation techniques for high-order elements show real benefits with respect to their original alternatives. Although HDG requires more computational resources, it is important to remark that it also provides a higher order of accuracy than CG and HCG methods.