Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods☆
Introduction
The numerical propagation of waves poses a significant challenge in scientific computation. Many alternative approaches have been explored in the quest for a stable method that can efficiently resolve the wave without excessive dissipation or dispersion, particularly in the context of high frequency applications. Some of the more promising domain based approaches involve the use of higher order schemes including spectral element methods [8], [10], higher order standard Galerkin finite element methods [3], [17], [25] and, more recently, higher order discontinuous Galerkin finite element methods [2], [4], [5], [6], [12], [13], [27].
The study of the dispersive and dissipative properties of a method provides insight into the ability of a method to accurately propagate a wave. Indeed, the order of accuracy of the discrete dispersion relation is often used as a basis for ranking different methods.
Higher order standard Galerkin finite element schemes for the Helmholtz equation in one space dimension were studied by Thompson and Pinsky [25] and Ihlenburg and Babuška [17], [18]. Recently [1], sharp estimates were obtained for the dispersive behaviour of higher order elements for the Helmholtz equation in multi-dimensions using tensor product elements.
The dispersive properties of higher order discontinuous Galerkin finite element methods have been studied in [14], [15], [16], [23]. In particular, Hu and Atkins [14] examine the dispersion properties of the approximation of the scalar advection equation in one dimension in the limit hk→0, for methods of order up to 16 using a computer algebra approach. On the basis of the computations, it was conjectured that the discrete wave-numbers are related to certain Padé approximants and that the dispersion relation for an Nth order method is accurate to order 2N+3 in hk for the dispersion error and order 2N+2 for the dissipation error. These orders of accuracy exceed those for the standard Galerkin finite element procedure [25].
The present work is concerned with the analysis of the dispersive behaviour of high order discontinuous Galerkin finite element methods. One by-product is a proof of correctness of the conjectures of Hu and Atkins (see Theorem 2). Moreover, Theorem 2 gives the coefficient of the leading terms in the error which, in view of the fact that in practical computations hk is finite, may be viewed as being of at least as much practical relevance as the order of decay. It is found that the leading coefficient decreases rapidly with increasing order N suggesting it may be advantageous to increase the order N whilst maintaining a fixed mesh.
This idea is pursued in Theorem 3 where it is shown that as the order N is increased, the dissipation and dispersion errors pass through three different phases depending on the size of N relative to hk. In the unresolved regime where 2N+1<hk−o(hk)1/3, the error oscillates without decay as the order is increased. At the opposite extreme, if the order is large, specifically 2N+1>hk+o(hk)1/3, then the error reduces at a super-exponential rate. The error decreases at an algebraic rate in the transition zone between these extremes.
The super-exponential rate of convergence in the resolved regime, where 2N+1>hk+o(hk)1/3, means that it is unnecessary to increase the order N much beyond this threshold. Instead, a practical alternative consists of tracking the envelope where the super-exponential phase begins by choosing the order of approximation so that 2N+1≈κhk for some fixed constant κ>1. In Theorem 4, we prove that this approach results in an exponential accurate discrete dispersion relation.
It is illuminating to compare these results with those for the continuous Galerkin finite element method analysed in [1]. The nature and the analysis of the discrete dispersion relation is quite different in the present situation, and this is reflected by the fact that the discontinuous Galerkin method has a higher order of accuracy in the limit hk→0. On the other hand, in the limit as N→∞, the threshold where the method resolves the wave is identical to that for the continuous Galerkin method despite the fact that the argument is completely different. This means that the better dispersive behaviour of the discontinuous Galerkin method in the limit hk→0 fails to carry through to the limit N→∞.
The remainder of this paper is organised as follows. We begin by describing the model problem and the details of the discontinuous Galerkin discretisation, and then give a detailed description of the theoretical results along with supporting numerical evidence. Section 3 is devoted to the study of the errors in certain types of Padé approximants of the exponential with particular attention to the situation where the order of the approximant is comparable to the argument, and where both are large. The link between the dispersive behaviour and the Padé approximants is established in the following section where we study a certain eigenvalue problem. We conclude with the proofs of the results stated in Section 2.
Section snippets
Model problem
Consider the linear advection equation in , ,subject to appropriate initial conditions. The advective field is assumed constant and we orient our Cartesian coordinate system so that α has non-negative components. It is well-known that this equation admits non-trivial solutions of the formwhere ω is a prescribed frequency and is the corresponding wave-vector. Inserting this expression into Eq. (1) and simplifying shows that the equation admits a
Padé approximant to the exponential
The study of Padé approximants of the exponential ez has enjoyed a long history going back to the original work of Padé himself [22] where the following results, quoted from Varga [26], are obtained for non-negative integers p and q:with remainder given byHere, denotes the confluent hypergeometric function defined by the seriesor, if we adopt
Analysis of an eigenvalue problem
Properties of the following eigenvalue problem will prove useful in the analysis of the dispersion error:
Find and such that for given ,As usual, the condition under which the eigenvalue problem will possess non-trivial solutions reduces to an algebraic equation for the eigenvalue λ, which we now proceed to identify.
Proofs of main results
Finally, we present the proofs of the results described in Section 2.3.
Acknowledgements
Support for this work from the Leverhulme Trust through a Leverhulme Trust Fellowship is gratefully acknowledged. This work was completed while the author was visiting the Newton Institute for Mathematical Sciences, Cambridge, UK.
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This paper is dedicated to Donald Kershaw on occasion of his seventy fifth birthday.