Elementi finiti discontinuous Galerkin ibridizzabili per problemi ellittici in 3D

The aim of this Master thesis work is a detailed analysis of Hybridizable Discontinuous Galerkin methods (HDG) for convection-diffusion-reaction equations and its implementation via object-oriented and Octave programming. The thesis is organized in six chapters whose contents are shortly described below. Chapter 1 focuses on the importance of numerical approximation of partial differential equations; in addition, particular attention is paid to the conceptual path to be followed in the resolution of a problem of physical interest. Chapter 2 analyizes in depth Discontinuous Galerkin (DG) methods, in terms of age of the method, size of problem, meaning of degrees of freedom, and other important features. A particular class of DG methods, called hybridizable DG methods, is then studied in detail. Chapters 3 and 4 are devoted to a thorough description of, respectively, C++ classes and Octave code, written for solving convection-diffusion-reaction equations using HDG methods. Chapter 5 is divided in two parts: the first one deals with experimental rate of convergence of L2 errors for five different unknowns; the second one is devoted to simulate a Darcy flow in a porous bulk with heterogeneous permeability. Chapter 6 sums up the main ideas developed in the thesis and indicates potential extensions. Appendix A focuses on analytical and geometrical aspects (reference element, Piola transformation, post-processing, and mesh generation) non directly explained in the previous chapters. Appendix B is a brief tutorial to be read before using the code.