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Explicit discrete dispersion relations for the acoustic wave equation in d-dimensions using finite element, spectral element and optimally blended schemes

Ainsworth, Mark and Wajid, Hafiz Abdul (2010) Explicit discrete dispersion relations for the acoustic wave equation in d-dimensions using finite element, spectral element and optimally blended schemes. In: Computer Methods in Mathematics. Computer Methods in Mathematics: Advanced Structured Materials, 1 (1). Springer, pp. 3-17.

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Abstract

We study the dispersive properties of the acoustic wave equation for finite element, spectral element and optimally blended schemes using tensor product elements defined on rectangular grid in d-dimensions. We prove and give analytical expressions for the discrete dispersion relations for the above mentioned schemes. We find that for a rectangular grid (a) the analytical expressions for the discrete dispersion error in higher dimensions can be obtained using one dimensional discrete dispersion error expressions; (b) the optimum value of the blending parameter is p/(p + 1) for all p ∈ ℕ and for any number of spatial dimensions; (c) the optimal scheme guarantees two additional orders of accuracy compared with both finite and spectral element schemes; and (d) the absolute accuracy of the optimally blended scheme is and times better than that of the pure finite and spectral element schemes respectively.