IAM770 - Special Topics: Discontinuous Galerkin Methods
Instructor(s): Hamdullah Yücel
Prerequisites: Consent of Instructor(s)
Course Catalogue Description
DG in One Spatial Dimension: linear system, implementation in MATLAB; Higher Dimensional Elliptic Problems: interior penalty methods, variational formulation, a priori error estimates, implementation in MATLAB, local discontinuous Galerkin method; DG for Convection Diffusion Problems: upwind scheme; Construction of Finite Element Spaces: Lagrange, Hermite, etc.; A Posteriori Error Analysis: residual-based, goal-oriented, hierarchical, equilibrated error estimators; Hybrid Discontinuous Galerkin Methods.
The aim of the course is to give the students an introduction to discontinuous Galerkin methods for solving problems in the engineering and the sciences described by systems of partial differential equations. These methods, most appropriately considered as a combination of finite volume and finite element methods, have become widely used during the last decade as a powerful tool for the simulation of challenging problems in the sciences and engineering.
The course covers both an overview of the theoretical properties of the methods, their efficient implementation, and more applied problems related to the multi-dimensional problems, an a posteriori error analysis, and adaptive refinement, illustrated using Matlab. We shall draw on application examples and illustrations from fluid dynamics but the focus on the course is on understanding the methods in sufficient depth to apply them to a broad range of problems.
This course is designed for graduate students majoring in mathematics as well as mathematically inclined graduate engineering students. At the end of this course, the student will:
- understand the mathematics behind discontinuous Galerkin methods: formulations, assembly for implementations, discrete spaces, approximation theory, error estimates;
- implement adaptive mesh refinement using various a posteriori error estimates;
- develop proficiency in the applications of the discontinuous Galerkin methods (modeling, analysis, and interpretation of results) to realistic engineering problems through the use of major commercial;
- general-purpose discontinuous Galerkin finite element code implement such methods and extensions in MATLAB or any programming language.
Course Learning Outcomes
Upon successful completion of this course, the student will be able to
- formulate and solve (with a computer) higher order partial differential equations in multi–dimensional problems using discontinuous Galerkin method;
- derive a priori and a posteriori error estimates using discontinuous Galerkin method;
- generate adaptive meshes using various a posteriori error estimates;
- apply the discontinuous Galerkin method in their thesis and understand the current research in this area;
- evaluate different techniques for solving problems and be able to motivate when to use existing software and when to write new code.
Tentative (Weekly) Outline
- Introduction and DG in One Spatial Dimension
- DG in One Spatial Dimension: linear System, implementation in MATLAB
- Higher Dimensional Elliptic Problems: interior penalty Methods, variational formulation
- Higher Dimensional Elliptic Problems: a priori error estimates
- Higher Dimensional Elliptic Problems: implementation in MATLAB
- Higher Dimensional Elliptic Problems: local discontinuous Galerkin method
- DG for Convection–Diffusion Problems: upwind Scheme
- Construction of Finite Element Spaces: Lagrange, Hermite, etc.
- A Posteriori Error Estimates: residual-based error estimator
- A Posteriori Error Estimates: goal-oriented and hierarchical estimator
- A Posteriori Error Estimates: equilibrated estimators
- Hybrid Discontinuous Galerkin Methods
- Direct Discontinuous Galerkin (DDG) Methods
- Review of the Topic Material
- Béatrice Riviére, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation, SIAM, 2008.
- D. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods, Springer 2012.
Supplementary Materials and ResourcesBook(s)
- J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin methods, Springer, 2008.