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
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.
Room 8500, TASC-2 Building (SFU).
This is the 8th PIMS-CSC Seminar in year 2009