## Recent Developments in Discontinuous Galerkin Finite Element Methods for PDEs

### May 9-11, 2012

## Main Speakers

**Franco Brezzi**, University of Pavia, Italy **Chi-Wang Shu**, Brown University

## Invited Speakers

**Slimane Adjerid**, Virginia Tech **Susanne Brenner**, Louisiana State University **Bernado Cockburn**, University of Minnesota **Clint Dawson**, University of Texas at Austin **Leszek Demkowicz**, University of Texas at Austin **Jean-Luc Guermond**, Texas A & M University **Charalambos Makridakis**, University of Crete, Greece **Donatella Marini**, University of Pavia, Italy **Ricardo Nochetto**, University of Maryland **Beatrice Riviere**, Rice University

## Abstracts

**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: \(\textbf{$C^0$}\)** 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.

**Organizing Committee**

Xiaobing Feng

Ohannes Karakashian

Yulong Xing

** Funded by:**The National Science Foundation

The Institute for Mathematics and its Applications

The University of Tennessee