BCAM Mini Course CFDCT: A brief introduction to high order Galerkin methods: spectral continuous and discontinuous Galerkin for fluid dynamics

A brief introduction to high order Galerkin methods: spectral continuous and discontinuous Galerkin for fluid dynamics

In the quest of exascale computing, the numerical approximation of partial differential equations should allow efficient parallelization. This requirement is even more important if high-order accuracy is required, as it can be observed by the difficulties of achieving ideal scalability on massively parallel computers if large-stencil approximations schemes are used. The impressive natural scalability on massively parallel computers as well as the low dissipation and dispersion error properties make high-order element based Galerkin (EBG) methods the perfect tool for high-fidelity simulations of fluid flows across a wide spectrum of applications that require high performance computing.

In this short course I will introduce you to the fundamentals of high-order continuous and discontinuous spectral elements and underline the major differences with respect to classical low order finite elements. I will provide the tools sufficient for a straightforward implementation as well as the details on how to take advantage of tensor product operations and inexact quadrature to reduce the operation count, hence making high-order EBG even faster than their low order counterpart.

In spite of the good properties underlined so far, the high order EBG approximation to non-linear equations is susceptible to Gibbs oscillation. This course will cover the fundamentals of de-aliasing and stabilization via classical filtering schemes and via dynamic artificial diffusion.

Because the construction of de-aliasing schemes for Galerkin methods has occupied most of my research since the time of my graduate studies at the Barcelona Supercomputing Center, I will also give an overview of my past and current work on this topic, with the goal of underlying in what directions researchers are moving today with respect to stabilization schemes. 

Programme:

10h-11h20 Basic concepts
11h20-11h40 coffe break
11h40-13h applications, with an overview of my research. 

Prerequisites: 

Basic knowledge of the finite element method.

Bibliography:

No bibliography is really necessary at this point. However, for those interested in digging further into this topic, I recommend the book by Karniadakis and Sherwin "Spectral/hp spectral element methods for computational fluid dynamics", Oxford U. Press (either 1st or 2nd editions).

Bio:

Simone Marras is a tenure track assistant professor in the Department of Mechanical and Industrial Engineering at the New Jersey Institute of Technology. He received an M.S. in Aerospace Engineering from Politecnico di Milano, Italy, and a Ph.D. from the Universitat Politècnica de Catalunya jointly with the Barcelona Supercomputing Center, Spain.

After receiving his doctorate, he spent two years at the department of Applied Mathematics at the Naval Postgraduate School as a National Research Council research associate (NRC of the National Academies of Sciences, Engineering, and Medicine) before moving to Stanford University for two years as a research scientist in the department of Geophysics. During the past ten years, Dr. Marras spent extended visits at UCLA, Cambridge University, the Naval Postgraduate School, and UT Austin.
His research interests include computational fluid dynamics for compressible flows, large eddy simulation of turbulence, and aeroacoustics via high-order Galerkin methods