 
Summary: Theory of the discontinuous Galerkin finite element method for
nonstationary convectiondiffusion problems and applications to
compressible flow1
Miloslav Feistauer,
Charles University Prague, Faculty of Mathematics and Physics
email: feist@karlin.mff.cuni.cz
ABSTRACT
In this lecture we shall be concerned with several aspects of the numerical solution of convection
diffusion problems by the discontinuous Galerkin finite element method (DGFEM) and applications
to compressible flow. The DGFEM is based on a piecewise polynomial approximation of the sought
solution without any requirement on the continuity on interfaces between neighbouring elements. It
is particularly convenient for the solution of conservation laws with discontinuous solutions or singu
larly perturbed convectiondiffusion problems with dominating convection, when solutions contain very
steep gradients.
The first subject is the analysis of error estimates of the DGFEM applied to the space semidiscretiza
tion of nonstationary convectiondiffusion problems. We shall discuss the error estimates in L2(H1)
and L(L2)norm for linear and nonlinear problems, their optimality and uniformity with respect to
the diffusion coefficient tending to zero. We shall also mention the spacetime DG discretization. The
theoretical results will be illustrated by numerical experiments.
In the second part, some applications of the DGFEM to the simulation of compressible flow, i.e. the so
