Analysis of the discontinuous Galerkin method applied to the diffusion operator

The two established formulations for the treatment of diffusion by a discontinuous Galerkin method of degree k do not converge at the optimal rate of k + 1 for all k. A variation on one of these methods that recovers the optimal rate of convergence is presented. This formulation is also more compact than the original form which makes it better suited for implementation in a parallel computing environment. Fourier analysis is performed for both the established approaches and the new method to determine the stability bounds for these methods when used in conjunction with RungeKutta time marching methods. Introduction The discontinuous Galerkin method has proven to be a very robust and efficient formulation for wave propagation.1-3 However, to be applicable to the broader class of problems encountered in fluid mechanics, a method must be able to treat viscous and thermal diffusion effects with equal accuracy. As with many finite-element methods, the formulation of the discontinuous Galerkin method for a convectiondiffusion equation requires careful attention in order to avoid inconsistencies. Two consistent formulations of the discontinuous Galerkin method applied to the diffusion operator are described in literature. The first formulation is procedurally similar to the “mixed” finite-element formulation commonly used in the finite-element community in that auxiliary variables are introduced to reduce the equation to a set of first order equations. However, when applied to the discontinuous Galerkin method, the spaces of the solution and the auxiliary variables ‘Senior Research Scientist, Computational Modeling and Simulation Branch tprofessor, Division of Applied Mathematics Copyright @ 1999 by the American Institute of Aeronautics and Astronautics, Inc. No copyright is asserted in the United States under Title 17, U.S. Code. The U.S. Government has a royalty-free license to exercise all rights under the copyright claimed herein for Governmental Purposes. All other rights are reserved by the copyright owner. are the same and an explicit check of the inf-sup condition is not required for stability. Also the auxiliary variables can be locally solved and eliminated without the need of any global solver. This approach was first applied to the two-dimensional Navier-Stokes equations by Bassi and Rebay4 and will be referred to as the BR formulation. The second approach, recently proposed by Oden, Babuska, and Baumanq5a6 is similar to a penalty method in that the formulation is stabilized by the addition of a term that is proportional to the jump in the solution between neighboring elements; however, the coefficient of the jump is a function of the solution gradient and is not strictly positive. This approach will be referred to as the OBB formulation. Each formulation has some advantages over the other, however, both methods share a common flaw in that neither converges at the optimal rate of k + 1 for all degree k. At first glance, the OBB formulation appears to require less storage and computational overhead than the BR formulation; however, in practice, the storage and computational requirements of the two methods are similar. The OBB formulation is more compact which makes it better suited for implementation on parallel computing platforms. Proofs of both stability and convergence for the discontinuous Galerkin method developed for convection also apply to the convection-diffusion case when the BR formulation is used.7 In reference 6, the rigorous proof for the stability of the OBB formulation, in terms of the infsup condition, is given for the one-dimensional steady case, and numerical evidence of stability is given for the multi-dimensional case. No convergence result is proven for the multidimensional steady case or for any time dependent case. We compare the stability, accuracy, and efficiency of these two formulations and introduce a variation of the BR formulation that converges at the optimal rate for all k. This modification also permits an efficient parallel implementation that incurs no storage overhead (over that required to solve advection equations). In addition, the modified method can be implemented on parallel computers with no additional communica-