## PIMS-CSC Seminar: A New Class of Discontinuous Petrov-Galerkin (DPG) Finite Element (FE) Methods

- Date: 09/18/2009

Leszek Demkowicz (University of Texas at Austin)

Jay Gopalakrishnan (University of Florida)

Simon Fraser University

A New Class of Discontinuous Petrov-Galerkin (DPG) Finite Element (FE) Methods for Convection-Dominated Diffusion and Compression Navier-Stokes Equations

The hp-adaptive finite elements combine elements of varying size h

and polynomial order p to deliver approximation properties superior

to any other discretization methods. The best approximation error

converges exponentially fast to zero as a function of number of

degrees-of-freedom. The hp methods are thus a natural candidate

for singularly perturbed problems experiencing internal

or boundary layers like in compressible gas dynamics.

This is the good news. The bad news is that only a small number

of variational formulations is stable for hp-discretizations.

By the hp-stability we mean a situation where the discretization

error can be bounded by the best approximation error times

a constant that is independent of both h and p. To this class

belong classical elliptic problems (linear and non-linear),

and a large class of wave propagation problems whose discretization

is based on hp spaces reproducing the classical exact grad-curl-div

sequence. Examples include acoustics, Maxwell, elastodynamics,

poroelasticity and various coupled and multiphysics problems.

We will present a new paradigm for constructing discretization

schemes for virtually arbitrary systems of linear PDE's that

remain stable for arbitrary hp meshes, extending thus dramatically

the applicability of hp approximations. For a start, we focus

on a challenging model problem - convection dominated diffusion.

The presented methodology incorporates the following features:

1. The problem of interest is formulated as a system of first

order PDE's in the distributional (weak) form, i.e. ALL derivatives

are moved to test functions. We use the DG setting, i.e. the

integration by parts is done over individual elements.

2. As a consequence, the unknowns include not only field variables within

elements but also fluxes on interelement boundaries. We DO NOT

use the concept of a numerical flux nurtured for the last 60 years

but, instead, treat the fluxes as independent, additional unknows.

3. For each trial function corresponding to either field or flux

variable, we determine a corresponding OPTIMAL test function

by solving an auxiliary LOCAL problem on one element. The use

of optimal test functions guarantees attaining the supremum

in the famous inf-sup condition from Babuska-Brezzi theory.

4. The local problems for determining optimal test functions

are solved approximately with an enhanced approximation (a locally

hp-refined mesh).

By selecting right norms for test functions, we can obtain amazing

stability properties uniform not only with respect to discretization

parameters but the diffusion constants as well.

We will present initial theoretical results and numerous numerical

experiments supporting the claims.

We are in process of extending the methodology to a class of steady-state

compressible Navier-Stokes equations with applications to transonic

and hypersonic flows. We will cocnlyde the presentation by

showing some preliminary numerical results for 1D Burgers and NS equations.

The presented research has been partially supported by Boeing, DOE and NSF.

2:30pm,

Room 8500, TASC-2 Building (SFU).

This is the 8th PIMS-CSC Seminar in year 2009