Discrete Dispersion Relation for hp-Version Finite Element Approximation at High Wave Number

The dispersive properties of high order finite element schemes are analyzed in the setting of the Helmholtz equation, and an explicit form of the discrete dispersion relation is obtained for elements of arbitrary order. It is shown that the numerical dispersion displays three different types of behavior, 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 behavior and rates of decay of the dispersion error in the differing regimes. All estimates are fully explicit and are shown to be sharp. Limits are obtained, where transitions between the different regimes occur, and used to provide guidelines for the selection of the mesh-size and the polynomial order in terms of the wave number so that the dispersion error is controlled.