433

RWTH Publication No: 540006 2015 IGPM433.pdf 
TITLE 
On the Stability of DPG Formulations of Transport Equations 
AUTHORS 
Dirk Broersen, Wolfgang Dahmen and Rob P. Steveson 
ABSTRACT 
In this paper we formulate and analyze a Discontinuous Petrov
Galerkin formulation of linear transport equations with variable convection
fields. We show that a corresponding infinite dimensional meshdependent
variational formulation, in which besides the principal field also its trace on
the mesh skeleton is an unknown, is uniformly stable with respect to the mesh,
where the test space is a certain product space over the underlying domain
partition.
Our main result states then the following. For piecewise polynomial trial
spaces of degree m, we show under mild assumptions on the convection field
that piecewise polynomial test spaces of degree m + 1 over a refinement of the
primal partition with uniformly bounded refinement depth give rise to uniformly (with respect to the mesh size) stable PetrovGalerkin discretizations.
The partitions are required to be shape regular but need not be quasiuniform.
An important startup ingredient is that for a constant convection field one can
identify the exact optimal test functions with respect to a suitably modified
but uniformly equivalent broken test space norm as piecewise polynomials.
These test functions are then varied towards simpler and stably computable
nearoptimal test functions for which the above result is derived via a perturbation analysis. We conclude indicating some consequences of the results that
will be treated in forthcoming work 
KEYWORDS 
Discontinuous Petrov Galerkinformulation of transport equations, optimal and nearoptimal test spaces, stability 
DOI 
10.1090/mcom/3242 