Author: Mark Ainsworth
Title of the contribution: Dispersive and Dissipative Behaviour of High Order Galerkin Schemes
Abstract:
We consider the dispersive properties of Galerkin finite element methods for wave propagation. The dispersive properties of conforming finite element scheme are analysed in the setting of the Helmholtz equation and an explicit form the discrete dispersion relation is obtained for elements of arbitrary order. It is shown that the numerical dispersion displays three different types of behaviour depending on the size of the order of the method relative to the mesh-size and the wave number. Quantitative estimates are obtained for the behaviour and rates of decay of the dispersion error in the differing regimes.
We then turn our attention to the dispersive properties of a high order discontinuous Galerkin scheme. In the small wave-number limit, we show the discontinuous Galerkin gives a higher order of accuracy than standard Galerkin. If the mesh-size is fixed and the order is increased, it is shown that the dissipation and dispersion errors decay at a super-exponential rate when the order is sufficiently large, and we give sharp bounds on where the transition occurs. Finally, we analyse the case of most practical interest where the order is chosen on the envelope where the super-exponential convergence sets in using sharp uniform asymptotics.
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