Consistent higher degree Petrov–Galerkin methods for the solution of the transient convection–diffusion equation

Abstract

Abstract The solution of the convection–diffusion equation for convection dominated problems is examined using both N + 1 and N + 2 degree Petrov–Galerkin finite element methods in space and a Crank–Nicolson finite difference scheme in time. While traditional N + 1 degree Petrov–Galerkin methods, which use test functions one polynomial degree higher than the trial functions, work well for steady‐state problems, they fail to adequately improve the solution for the transient problem. However, using novel N + 2 degree Petrov–Galerkin methods, which use test functions two polynomial degrees higher than the trial functions, yields dramatically improved solutions which in fact get better as the Courani number increases to 1·0. Specifically, cubic test functions with linear trial functions and quartic test functions in conjunction with quadratic trial functions are examined. Analysis and examples indicate that N + 2 degree Petrov–Galerkin methods very effectively eliminate space and especially time truncation errors. This results in substantially improved phase behaviour while not adversely affecting the ratio of numerical to analytical damping.