Error Estimates on Hybridizable Discontinuous Galerkin Methods for Parabolic Equations with Nonlinear Coefficients

Hyung Kyu Jun,Tay Suh,Minam Moon

doi:10.1155/2017/9736818

Abstract

HDG method has been widely used as an effective numerical technique to obtain physically relevant solutions for PDE. In a practical setting, PDE comes with nonlinear coefficients. Hence, it is inevitable to consider how to obtain an approximate solution for PDE with nonlinear coefficients. Research on using HDG method for PDE with nonlinear coefficients has been conducted along with results obtained from computer simulations. However, error analysis on HDG method for such settings has been limited. In this research, we give error estimations of the hybridizable discontinuous Galerkin (HDG) method for parabolic equations with nonlinear coefficients. We first review the classical HDG method and define notions that will be used throughout the paper. Then, we will give bounds for our estimates when nonlinear coefficients obey “Lipschitz” condition. We will then prove our main result that the errors for our estimations are bounded.

Highlights

In this paper, we obtain uniform-in-time convergence error estimates for the semidiscretization by hybridizable discontinuous Galerkin (HDG) methods for the parabolic equation with nonlinear coefficient.ut − ∇ ⋅ (κ (u) ∇u) = f, in Ω × (0, T], (1a)u (x, t) = 0, on ∂Ω × (0, T], (1b)u (x, 0) = u0, on Ω. (1c)Here, bounded κ(u) ≥ κ0 polyhedral> 0 is domain a nonlinear in Rn, n =coefficient, 2, 3, and TΩ is a is final time.Parabolic equation describes the distribution of heat in a certain region over time with given boundary conditions
Details regarding parabolic equations and the way of applying numerical approximations in various contexts can be found in diverse sources such as [1,2,3] or [4]
Despite the importance of solving parabolic equations and the substantial number of researches on applying various techniques to give approximate solutions that tend to hint that, at least for HDG method seems to be the best tools we have in our pocket, error analysis has not been conducted yet

Summary

Introduction

We obtain uniform-in-time convergence error estimates for the semidiscretization by hybridizable discontinuous Galerkin (HDG) methods for the parabolic equation with nonlinear coefficient. Various methods have been developed to find approximate solutions for given parabolic equations having nonlinear coefficients (or more generally, any PDE). Based on optimal convergence and superconvergence of HDG methods, local postprocessing was developed in [24] for linear convection-diffusion equations and in [25] for nonlinear case to increase the convergence order of numerical solutions. Despite the importance of solving parabolic equations and the substantial number of researches on applying various techniques to give approximate solutions that tend to hint that, at least for HDG method seems to be the best tools we have in our pocket, error analysis has not been conducted yet. We will give error estimations for the proposed HDG method

Preliminaries

A Priori Estimate

Error Estimations

Conclusion