**The 2012 John H. Barrett
Lectures Abstracts**

May 9-11, 2012, SERF 307

**Franco Brezzi**
(University of Pavia, Italy)

Title: **Theoretical aspects
of DG methods for stationary problems**:

Part 1. **Mathematical Background of
DG methods. **

The lecture will provide the
basic mathematical instruments for proving theorems (and for making wise
guesses) on DG methods. In spite of its theoretical target, there will be
rather few technicalities. The main inequalities will be proved in the
one-dimensional case, using Elementary
Calculus. Indeed, the idea will be to understand why they hold, rather
than memorizing the minimal assumptions or the detailed proofs. Then we will
see and analyze the simpler cases of DG discretizations
of Laplace equation.

Part 2. **Classical DG methods for elliptic
problems of order 2 and 4.**

The lecture will recall the
basic convergence results for the most common variants of DG methods for linear elliptic
second order problems, like Laplace, Navier, Stokes, Kirchhoff-Love. Here too
we will concentrate on the reasons why a method works, or doesn't work, rather
than on the detailed proof. Some still open questions will also be discussed.

Part 3. **Connections
between DG and other methods. **

It is by now clear that DG
methods have close connections with other Finite Element Methods, such as
Nonconforming FEM, Mixed FEM or Hybrid FEM. The connections are both in the use
of DG-variants
of the other methods, in the study of the limit for some penalty term going to
infinity, or just in some
reinterpretation of one method in terms of another. Here there is still ample
room for cross-fertilization
based research. A hint will also be given to the possible use of DG versions of
the (brand new) Virtual Element
Methods.

**Chi-Wang Shu** (Brown
University)

Title: **Discontinuous Galerkin methods for
time dependent problems: survey and recent developments, Part 1-3**

In these lectures
we will describe discontinuous Galerkin methods for
solving time dependent partial differential equations, including hyperbolic,
convection diffusion, and dispersive wave equations. Algorithm formulation,
stability analysis and error estimates, and efficient implementation issues
will be discussed. Recent developments including superconvergence,
positivity-preserving, and discontinuous Galerkin
methods for problems involving delta-function in their solutions will be
addressed.

**Slimane**** Adjerid** (Virginia Tech)

Title: **Accurate error estimates
and superconveregnce for DG methods**

We present several superconvergence results and asymptotically exact a
posteriori estimates for discontinuous Galerkin
methods applied to convection and convection-diffusion problems. We
perform an analysis of the local DG error to construct simple, efficient, and
asymptotically correct a posteriori error estimates for a minimal dissipation
LDG solutions of two-dimensional diffusion and convection-diffusion problems on
rectangular meshes. We also present new superconvergence
results with accurate error estimates for three-dimensional hyperbolic problems
on tetrahedral meshes. On each element, the asymptotic behavior of DG errors
depends on the mesh orientation with respect to the problem characteristics.
Thus, elements are classified according to the number of inflow and outflow
faces and in all cases enriched finite elements spaces are needed to show pointwise superconvergence.
Numerical results are presented to validate the theory.

**Susanne Brenner** (Louisiana State University)

Title: **C^0 Interior Penalty Methods**

C^0 interior
penalty methods are discontinuous Galerkin methods
for fourth order problems that use standard Lagrange finite element spaces for
second order problems. In this talk we will discuss convergence, adaptivity, fast solvers and applications of these methods.

**Bernardo Cockburn** (University of Minnesota)

Title: **Devising superconvergent DG methods**

We show
how to reduce the devising of superconvergent HDG methods
for diffusion to the verification of some simple properties relating the local
spaces defining the methods. We pont out that all the main mixed and the known superconvergent
HDG methods satisfy those properties. We then use these properties to constrcut new superconvergent
mixed and HDG methods. Finally, we show how, given any mixed or HDG superconvergent method for diffusion, we can constrcut superconvergent HDG
methods for Stokes, and two different formulations of the linear elasticity
system.

**Clint Dawson** (University of Texas at Austin)

Title: **Local time stepping in DG methods and applications to the
shallow water equations**

One of the
limitations of DG methods, especially on highly unstructured grids, is the CFL
limitation imposed by using explicit time stepping methods. In coastal
modeling applications, unstructured grids are required to handle complicated
coastlines and bathymetry. These grids are highly graded and can vary in
size by orders of magnitude. Using a globally constrained CFL time step
becomes expensive. Local time stepping methods, also known as multirate methods, are one way to circumvent this issue.
One of the difficulties in these methods is imposing conservation
constraints and maintaining accuracy. We will discuss one such approach
recently developed and applied to shallow water and overland flow models.

**Leszek**** Demkowicz** (University of Texas at Austin)

Title: **Discontinuous Petrov-Galerkin methods with optional test functions**

The talk will provide a high level overview of what has been
learned about the DPG method since our original series of contributions on the
subject in 2009 [1,2,3]. Here are a few highlights of the presentation.

1. The method falls into the class of minimum-residual
(least-squares) methods. What makes is different from classical least squares approach
is the fact that the residuals are computed in dual norms. This allows for
relaxation avoiding the overdiffusive behavior of
strong least squares (L2-valued) methods.

2. Use of the dual norms necessitates an approximate inversion of Riesz operators. The inversion is made feasible by the use
of discontinuous test functions (broken Sobolev
spaces). Of particular importance is the mesh dependent, ultraweak
variational formulation originating from a first
order Friedrichs-type system of PDEs [13].

3. If the operator corresponding to the strong L2-setting of the first-order
system is bounded below, so is the operator corresponding to the ultra-weak variational formulation with practically the same (and, therefore,
mesh-independent) inf-sup constant. In simple words,
the ultra-weak variational formulation enabling the
computation of optimal test functions is as good as any classical variational formulation [7,8,9,13].

4. If the error in computing optimal test functions is negligible,
the discrete problem automatically inherits the good stability properties from
the continuous one. The method delivers the best approximation in terms of the
energy norm, i.e. the residual. Contrary to standard methods, the method comes
with a built-in posteriori-error evaluator (not estimator) that enables adaptivity [3,12]. To illustrate the points above, I
will use the Stokes problem, explaining how the general theory specializes to
this case and illustrating it with several numerical experiments. The rest of
the talk is going to be a quick overview w/o going into details, just punchlines.

5. For singular-perturbation problems, one can construct in a
systematic way an appropriate test norm with which the method is robust, i.e.
the stability is uniform with respect to the perturbation parameter. Construction
reduces to the stability analysis for Friedrichs
systems in the strong L^2 setting [12].

6. There are more than one way to construct such optimal test
norms but not all good test norms are equally feasible. Some optimal test norms
may deliver optimal test functions with boundary layers whose resolution may be
as difficult as the solution of the original problem [12].

7. Accounting for the approximation of optimal test functions is
possible (although not easy).
Surprisingly or not, we end up using elements of Brezzi's
theory for mixed problems [8].

8. A robust (pollution-free) discretization of wave propagation
problems stands out on its own as an example of a problem where one benefits by
minimizing discrete rather than continuous residual.

Some of the open problems include:

1. The method is still in its infancy for non-linear problems
[5].

2. A general, automatic hp-adaptivity is
still in a planning stage [3,9,12].

3. The essence of the strong stability analysis seems frequently
to lie in selection of boundary conditions [12].

4. Construction of preconditioners and
general iterative schemes has just begun.

[1] L. Demkowicz and J. Gopalakrishnan, ``A class of discontinuous Petrov-Galerkin methods. PartI: The transport equation,'' CMAME: 199,
23-24, 1558--1572, 2010.

[2] L. Demkowicz and J. Gopalakrishnan,
``A class of discontinuous Petrov-Galerkin methods.
Part II: Optimal test functions,''
Num. Meth. Part. D.E.:27, 70-105, 2011.

[3] L. Demkowicz, J. Gopalakrishnan
and A. Niemi, ``A class of discontinuous Petrov-Galerkin methods. Part III: Adaptivity,''
App. Num Math., in
print.

[4] A. Niemi, J. Bramwell
and L. Demkowicz, ``Discontinuous Petrov-Galerkin
Method with Optimal Test Functions for Thin-Body Problems in Solid Mechanics,''
CMAME: 200,

1291-1300, 2011.

[5] J. Zitelli, I.
Muga, L, Demkowicz, J. Gopalakrishnan,
D. Pardo and V. Calo, ``A
class of discontinuous Petrov- Galerkin
methods. IV: Wave propagation problems,''

J.Comp. Phys.:
230, 2406-2432, 2011.

[6] J. Chan, L. Demkowicz,
R. Moser and N Roberts, ``A class of discontinuous Petrov-Galerkin
methods. Part V: Solution of 1D Burgers and Navier--Stokes
Equations,''

ICES
Report 2010-25.

[7] L. Demkowicz and J. Gopalakrishnan, ``Analysis of the DPG Method for the
Poisson Equation,'' SIAM J. Num. Anal., 2011.

[8] L. Demkowicz, J. Gopalakrishnan,
I. Muga, and J. Zitelli.
``Wavenumber Explicit Analysis for a DPG Method for the Multidimensional
Helmholtz Equation'',

CMAME, in print.

[9] J. Gopalakrishnan and W. Qiu, ``An Analysis of the Practical DPG Method'', submitted
to Num. Math.

[10 J. Bramwell, L. Demkowicz,
J. Gopalakrishnan, and W. Qiu.
``A Locking-free hp DPG
Method for Linear Elasticity with Symmetric Stresses'', Technical Report

2369,
Institute for Mathematics and Its
Applications, May 2011,
(http://www.ima.umn.edu/preprints/may2011/may2011.html)}.

[11] T. Bui-Thanh, L. Demkowicz and O. Ghattas,
``Constructively Well-Posed Approximation Methods with Unity Inf-Sup and Continuity'', Math. Comp. 2012,
accepted.

[12] L. Demkowicz and
J. Li, ``Numerical Simulations of Cloaking Problems using a DPG Method'', ICES
Report 2011/31.

[13] L. Demkowicz and M. Heuer, ``Robust DPG Method for Convection-Dominated
Diffusion Problems'', ICES Report 2011/33.

[14] T. Bui-Thanh, L. Demkowicz and O. Ghattas, ``A
Unified Discontinuous Petrov-Galerkin Method and its Analysis for Friedrichs'
Systems'', ICES Report 2011/34.

[15] T. Bui-Thanh, L. Demkowicz and O. Ghattas, ``A
Relation between the Discontinuous Petrov--Galerkin Method and the Discontinuous Galerkin Method'', ICES

Report
2011/45.

**Jean-Luc Guermond** (Texas A
& M University)

Title: **Discontinuous Galerkin
methods for the radiative transport equation**

** **

We introduce a new
discontinuous Galerkin (DG) method with weighted stabilization
for the linear Boltzmann equation applied to particle transport. The asymptotic
analysis demonstrates that the new formulation does not suffer from the limitations
of standard upwind methods in the thick diffusive regime; in particular, the
new method yields the correct diffusion limit for any approximation order,
including piecewise constant discontinuous finite elements. Numerical tests on
well-established benchmark problems demonstrate the superiority of the new
method. The improvement is particularly significant when employing piecewise
constant DG approximation for which standard upwinding
is known to perform poorly in the thick diffusion
limit.

**Charalambos**** Makridakis** (University of Crete, Greece)

Title: **Transport, dispersion and local reconstructions in discontinuous
Galerkin methods**

We discuss DG
methods for transport and higher-order PDEs describing dispersion/capillarity
effects. These equations arise not only as models for solitary waves but
also in multiscale modeling and in phase
transitions. In particular we shall consider the isothermal *Navier**-Stokes Korteweg *system
for which we present thermodynamically consistent DG schemes. We discuss
issues related to the error analysis of the approximations. We present recent
results related to a posteriori error control of DG methods for linear
hyperbolic and dispersive equations by utilizing appropriate local reconstructions.

**Donatella**** Marini **(University
of Pavia, Italy)

Title: **Virtual elements and DG**

Virtual
Element Methods (VEM) are the latest evolution of the
Mimetic Finite Difference Method, and can be considered to be more close to the
Finite Element approach. They combine the ductility of mimetic finite
differences for dealing with rather weird element geometries with the
simplicity of implementation of Finite Elements. Moreover they make it possible
to construct quite easily high-order and high-regularity approximations (and in
this respect they represent a significant improvement with respect to both FE
and MFD methods). In the present
talk we will first introduce the general idea of continuous VEM, underlying the
similarities with classical finite elements. Then we will consider their
extension to discontinuous formulations.

**Ricardo Nochetto**
(University of Maryland)

Title: **Time-discrete higher order ALE formulations: a dG approach**

Arbitrary Lagrangian Eulerian (ALE)
formulations deal with PDEs on deformable domains upon extending the domain
velocity from the boundary into the bulk with the purpose of keeping mesh
regularity. This arbitrary extension has no effect on the stability of the PDE
but may influence that of a discrete scheme. In fact, the only scheme that is
provably unconditionally stable is the Euler method. We propose time-discrete
discontinuous Galerkin (dG)
numerical schemes of any order for a time-dependent advection-diffusion model
problem in moving domains, study their stability properties, and derive optimal
a priori and a posteriori error estimates. The analysis hinges on the validity
of the Reynolds' identity for dG and exploits the variational structure of dG. We
also study the effect of quadrature and the practical Runge-Kutta-Radau
(RKR) methods of any order. This is joint work with A. Bonito and I. Kyza.

**Beatrice Riviere** (Rice
University)

Title: **Coupled free flows and porous media flows**

Mathematical and
numerical modeling of coupled Navier-Stokes (or
Stokes) and Darcy flows is a topic of growing interest. Applications include
the environmental problem of groundwater contamination through rivers, the
problem of flows through vuggy or fractured porous
media, the industrial manufacturing of filters, and the biological modeling of
the coupled circulatory system with the surrounding tissue. The most widely
used coupling model is based on the Beavers–Joseph–Saffman interface conditions.

In this lecture,
the mathematical analysis of the coupled equations is first briefly
presented. Various numerical discretizations are
then formulated: they employ discontinuous Galerkin
methods, finite element methods and mixed element methods. We show
that the choice of a particular discretization influences
the treatment of the interface conditions between the porous medium and
the free flow region. A priori error estimates are
derived. Finally, to model groundwater contamination,
the oupled flow problem is
combined with a transport equation. A discontinuous Galerkin
method is used to approximate the concentration of the contaminant.

** **

** **