Comparision of continuous and discontinuous Galerkin finite element methods for parabolic partial differential equations with implicit time stepping

Abstract

A number of different discretization techniques and algorithms have been developed for approximating the solution of parabolic partial differential equations. A standard approach, especially for applications that involve complex geometries, is the classic continuous Galerkin finite element method. This approach has a strong theoretical foundation and has been widely and successfully applied to this category of differential equations. One challenging sub-category of problems, however, are equations that include an advection term that is large relative to the second-order, diffusive term. For these advection dominated problems, the continuous Galerkin finite element method can become unstable and yield highly inaccurate results. An alternative to the continuous Galerkin finite element method is the discontinuous Galerkin finite element method, and, through the use of a numerical flux term used in deriving the weak form, the discontinuous approach has the potential to be much more stable in highly advective problems. However, the discontinuous Galerkin finite element method also has significantly more degrees-of-freedom due to the replication of nodes along element edges and vertices. The work presented here compares the computational cost, stability, and accuracy (when possible) of continuous and discontinuous Galerkin finite element methods for four different test problems, including the advection-diffusion equation, viscous Burgers' equation, and the Turing pattern formation equation system. The comparison is performed using as much shared code as possible between the two algorithms and direct, iterative, and multilevel linear solvers. The results show that, for implicit time stepping, the continuous Galerkin finite element method is typically 5-20 times less computationally expensive than the discontinuous Galerkin finite element method using the same finite element mesh and element order. However, the discontinuous Galerkin finite element method is significantly more stable than the continuous Galerkin finite element method for advection dominated problems and is able to obtain accurate approximate solutions for cases where the classic, un-stabilized continuous Galerkin finite element method fails.