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 inclinedgraduate 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;
• implement such methods and extensions in MATLAB or any programming language.