A High-Order HDG Method with Dubiner Basis for Elliptic Flow Problems

Manuela Bastidas,Bibiana Lopez-Rodríguez,Mauricio Osorio

doi:10.17230/ingciencia.16.32.2

Abstract

We propose a standard hybridizable discontinuous Galerkin (HDG) method to solve a classic problem in fluids mechanics: Darcy’s law. This model describes the behavior of a fluid trough a porous medium and it is relevant to the flow patterns on the macro scale. Here we present the theoretical results of existence and uniqueness of the weak and discontinuous solution of the second order elliptic equation, as well as the predicted convergence order of the HDG method. We highlight the use and implementation of Dubiner polynomial basis functions that allow us to develop a general and efficient high order numerical approximation. We also show some numerical examples that validate the theoretical results.

Highlights

The study of flow processes on porous media has gained particular strength during last decades
That we have proved that Problem P2G has a unique solution, we can introduce the main advantage of the hybridizable discontinuous Galerkin (HDG) method and reduce Problem P2G to a problem only over the skeleton
We have introduced the hibridizable discontinuous Galerking method (HDG) as a suitable idea for constructing high order approximations to the solution of Darcy’s law

Summary

Introduction

The study of flow processes on porous media has gained particular strength during last decades. We refer to [8] for a general overview of mixed methods applied to Darcy’s law This area has been widely studied from all fronts; many works during the last 20 years have shown progress in the numerical approximation of the Darcy equation and other equations of fluid mechanics using mainly the finite volume method and finite differences. We highlight some relevant results to solve the Stokes equation and other fluid problems using HDG, in [19],[20],[21],[22] Many of these papers lack a complete error analysis or only consider the case of constant permeability.

The hybridizable discontinuous Galerkin method

The local solvers

Error analysis

Dubiner basis

Numerical examples

Conclusions