Numerical studies of Galerkin-type time discretizations applied to transient convection-diffusion-reaction equations

Dr. Naveed Ahmed

July 24, 2012, 3:30 p.m. MT 132

It is well known that the Galerkin finite element method is unstable for the nu-
merical solution of convection-dominated problems since the solution is typically
polluted by spurious oscillations. To enhance the stability while keeping the ac-
curacy of the Galerkin method, several stabilization techniques have been devel-
oped. We consider the streamline upwind Petrov-Galerkin (SUPG) method and
local projection stabilization (LPS) method to stabilize the Galerkin discretiza-
tion. The SUPG formulation is strongly consistent whereas, the LPS method lies
in the class of symmetric stabilizations and weakly consistent.
We shall employ the combination of SUPG and LPS methods in space with the
variational type time-discretization schemes for the numerical solution of time-
dependent convection-diffusion-reaction equations. In particular, we consider the
discontinuous Galerkin (dG) and continuous Galerkin-Petrov (cGP) method to
discretize the problem in time. Several numerical tests have been performed to
assess the accuracy of the higher order time-discretization schemes. For smooth
solution, optimal order of convergence for cGP and dG-methods are obtained.
Further, numerical comparison of SUPG and LPS methods for the smooth so-
lution shows that both stabilization techniques perform quite similar and no
difference among them can be appreciated. Finally, the dependence of the results
on the stabilization parameters are discussed.