## Nonconforming and Discontinuous Galerkin Methods

### Motivation

Nonconforming and Discontinuous Galerkin methods are popular techniques for the numerical solutions of partial differential equations. In the context of finite elements methods, they generalize the well-known concept of conforming Galerkin methods and offer less "rigid" discrete trial and test functions. This may be used to enhance approximation and stability as well as to accommodate for physical properties of the discrete solution with relatively few degrees of freedom.

The analysis of both techniques is complicated by the fact that the discrete trial and test spaces are not conforming with (i.e., are not contained in) the continuous trial and test spaces. In particular, this so-called variational crime entails that, up to now, the theoretical results on these methods are less complete than their counterparts for conforming methods.

The workshop intends to assess and compare recent results on both techniques as well as generalizations of them, looking for a general theory.

### Participation

Participation is free. If you plan to attend, please send an email to pietro.zanotti@unimi.it not later than January 24, 2017.

### Schedule

Friday, January 27, 2017, 9:00-18:00
Sala di Rappresentanza del Dipartimento di Matematica, Via Saldini 50, 20133 Milano

09:00-09:05 Opening
09:05-09:50 F. Brezzi - DG and VEMS
09:55-10:40 P. Antonietti - Numerical modelling of seismic waves by high-order discontinuous methods

10:40-11:00 Coffee break in Aula C

12:35-14:30 Lunch break

16:00-16:25 Coffee break in Aula C

Andreas Veeser
Pietro Zanotti

### Support

The workshop is supported by
• PRIN 2012 "Innovative methodologies for PDE-based numerical modelling" coordinated by C. Canuto
• Dipartimento di Matematica dell'Università degli Studi di Milano

### Abstracts

(in alphabetic order with respect to the speakers)

Numerical modelling of seismic waves by high-order discontinuous methods
Paola Antonietti (MOX, Politecnico di Milano, Italy)

In this talk we present and analyse high-order discontinuous Galerkin (DG) methods for the numerical modelling of the ground motion induced by large earthquakes. DG methods allow for local adaptivity on discretization parameters, thus improving the quality of the solution without affecting the computational costs. This approach is particularly well suited for the simulation of complex wave phenomena, such as the seismic response of sedimentary basins or soil-structure interaction problems, where flexibility and accuracy are crucial in order to simulate correctly the wave-front field while keeping affordable the computational effort. The theoretical properties of the semidiscrete formulation are discussed including stability and error estimates. A discussion on the dissipation, dispersion and stability properties of the fully-discrete formulation obtained through an explicit time marching scheme is also presented. Some validation benchmarks are shown to verify the practical performance of the proposed scheme. We also present simulations of real large-scale seismic events in three-dimensional complex media that include both far-field to near-field as well as soil-structure interaction effects. The numerical results have been obtained with the high performance, open-source numerical code SPEED (https://speed.mox.polimi.it) jointly developed at Politecnico di Milano by The Laboratory for Modeling and Scientific Computing MOX of the Department of Mathematics and by the Department of Civil and Environmental Engineering.

DG and VEMs
Franco Brezzi (IMATI-CNR, Pavia, Italy)

The talk will discuss the roads connecting Discontinuous Galerkin and Virtual Element Methods (together with minor excursions on other methods suited for polytopal decompositions). We will recall the very basic features of Virtual Elements, and in particular their Conforming, Non-conforming and Discontinuous versions, together with the different stabilizing terms and the Serendipity variants.

Nonconforming error analysis
Alexandre Ern (Cermics, ENPC, Paris, France)

We present an extension of Strang's First Lemma that covers nonconforming approximation settings. We propose a possible way to formulate the boundedness and consistency properties so as to obtain an error estimate that bounds the error by the best-approximation error of the exact solution by just invoking a slight regularity assumption on the exact solution. We compare the obtained error estimate to that delivered by Strang's Second Lemma and we discuss the application to elliptic PDEs and to the stabilized approximation of first-order PDEs.

This is joint work with Jean-Luc Guermond.

On a priori error estimates in nonconforming eigenvalue discretization
Dietmar Gallistl (Karlsruhe Institute of Technology, Germany)

Nonconforming finite element methods (FEMs) exhibit several advantages such as stable low-order discretizations for the Stokes flow or simple schemes in the discretization of fourth-order PDEs such as the Kirchhoff-Love plate equation. The computation of guaranteed lower eigenvalue bounds, which is relevant in practical applications makes nonconforming schemes very attractive. The simplest instances of nonconforming FEMs are the nonconforming P1 FEM (also known as Crouzeix-Raviart FEM) for second-order problems and the Morley FEM for fourth-order problems.

This contribution presents a novel technique based on intermediate eigenvalue problems in the sum of the energy space and the discrete nonconforming finite element space. It leads to new a priori error estimates for the eigenvalue error $|\lambda - \lambda_h|$ in terms of the principal angle between the exact and the discrete invariant subspaces when multiple eigenvalues or clusters are approximated with nonconforming schemes.

Patch-wise local projection methods for convection-diffusion problems
Thirupathi Gudi (IISc Bangalore, India)

It is well known that the numerical solution of the convection dominated diffusion problem by the standard Galerkin method exhibits spurious oscillations. There were different remedies proposed in the literature to suppress those oscillations and further to enhance the accuracy of the solution by improved order of convergence. Examples are streamline diffusion method, local projection method, bubble stabilization technique, least square method and discontinuous Galerkin method, etc. In this talk, we will discuss on two new methods based on patch-wise local projection methods by using the conforming finite element space and the nonconforming finite element space, respectively. These methods are defined on a single mesh and without enriching the finite element space. Some numerical examples illustrating the theoretical results will be presented.

This is a joint work with Dr. Asha K. Dond.

Asymptotic preserving discontinuous Galerkin methods for the linear Boltzmann equation
Guido Kanschat (University of Heidelberg, Germany)

Discontinuous Galerkin methods were originally developed to solve the linear Boltzmann equation. But while they had been developed and analyzed in the 1970s and 80s with applications in radiative transfer and neutron transport in mind, it was pointed out later in the nuclear engineering community, that the upwind DG discretization by Reed and Hill may fail to produce physically relevant approximations, if the scattering mean free path length is smaller than the mesh size. Mathematical analysis reveals, that in this case, convergence is only achieved in a continuous subspace of the finite element space. By choosing a weighted upwinding, the conditions on the diffusion limit can be weakened; by choosing the stabilization carefully, the DG method can yield either the LDG method or the method by Ern and Guermond in its diffusion limit.

On the L2 stability of the discontinuous Galerkin elliptic projection
Charalambos Makridakis (University of Sussex, UK)

We discuss the stability properties of the discontinuous Galerkin elliptic projection in mesh dependent L2 norms in unstructured meshes. As a result we show new a priori error estimates in L2.

Quasi-optimality of nonconforming methods for linear variational problems
Pietro Zanotti (Università degli Studi di Milano, Italy)

We consider general nonconforming methods for linear elliptic variational problems, symmetric for simplicity. After the introduction of the abstract framework, we characterize their quasi-optimality in terms of suitable notions of stability and consistency. Moreover, we identify the quasi-optimality constant and discuss its ingredients. We conclude by applying these results to the construction of some nonconforming and DG methods.

This is a joint work with Andreas Veeser.

Last update: 20/01/2017 -- A. Veeser