Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications

by Jan S. Hesthaven, Tim Warburton

This book offers an introduction to the key ideas, basic analysis, and efficient implementation of discontinuous Galerkin finite element methods (DG-FEM) for the solution of partial differential equations. It covers all key theoretical results, including an overview of relevant results from approximation theory, convergence theory for numerical PDE’s, and orthogonal polynomials. Through embedded Matlab codes, coverage discusses and implements the algorithms for a number of classic systems of PDE’s: Maxwell’s equations, Euler equations, incompressible Navier-Stokes equations, and Poisson- and Helmholtz equations.

516 pages, Hardcover

First published December 18, 2007

Reviews

[Mike Rosing · Rating 1 out of 5]

This describes the 1D analysis of Nodal methods in great detail. The methods assume orthonormal functions which is straight forward in 1D. When they jump to 2D and 3D they give formulas for functions on triangles and tetrahedra. After many years of not understanding why the formulas don't work, I found that the functions given in the book are not actually orthogonal. I think this went unnoticed because the book says "we define the expansion coefficients through an interpolation to avoid problems with the need to evaluate multidimensional integrals." Using GPU's it is easy to compute 3D integrals, and show that the given functions are not orthogonal.

Even with orthogonal functions (which are not much different than given in the book) the method still does not give reasonable solutions. I am not sure why this is the case because the 1D analysis seems fairly good. For 3D problems, this method does not work.