Applied Math Seminar: A High-Order Discontinuous Galerkin Method for Variable-Coefficient Advection-Diffusion Problems

Seminar | November 8 | 4-5 p.m. | 891 Evans Hall

 Raunak Borker, Stanford University

 Department of Mathematics

A high-order discontinuous Galerkin method with Lagrange multipliers (DGLM) is presented for the solution of the steady advection-diffusion equation with a variable advection field in the high Peclet number regime. In this regime, the standard finite element method (FEM) might produce non-physical oscillations in the solution at practical mesh resolutions. Like a Discontinuous Enrichment Method (DEM) [1], the DGLM method described in this presentation overcomes the issue of spurious oscillations near boundary layers by attempting to resolve them using appropriate shape functions. Specifically, these are chosen as polynomials that are additively enriched with free-space solutions of the governing differential equation. Also like a DEM, the DGLM method presented here in enforces a weak continuity of the solution across inter-element boundaries using Lagrange multipliers and is amenable to efficient numerical technique of static condensation. DGLM is based on the primal hybrid-variational formulation. Unlike a Hybrid Discontinuous Galerkin (HDG) method [2] however, it operates directly on the second-order form of the advection-diffusion equation and does not require any stabilization.

Recently in [3], DGLM was developed for the constant coefficient advection-diffusion problem. The present work focuses on the extensions of this approach to the variable coefficient problem. The major challenge in this extension is in deriving free-space solutions for the variable advection-field case. This challenge is overcome by using asymptotic analysis which provides enrichment functions for exponential and parabolic boundary layers. The resulting DGLM method is theoretically proven to be well-posed.

This talk will introduce the DGLM framework, the enrichment functions for the constant and variable coefficient cases, and discuss possible extensions to the incompressible Navier-Stokes equations. Numerical results for the advection-diffusion equation demonstrating the benefits of the enrichment functions in overcoming oscillations by reducing the error constant while maintaining a high-order convergence will be presented.

[1] C. Farhat, I. Harari, L. P. Franca, The discontinuous enrichment method, Comput. Methods Appl. Mech. Eng., 190, 6455- 6479, 2001.

[2] N. C. Nguyen, J. Peraire, B. Cockburn, An implicit high-order hybridizable discontinuous Galerkin method for linear convection-diffusion equations, J. Comput. Phys., 228, 3232-3254, 2009.

[3] R. Borker, C. Farhat, R. Tezaur, A high-order discontinuous Galerkin method for unsteady advection-diffusion problems, J. Comput. Phys., 332, 2017.