A novel spherical polar finite element for the solution of the steady‐state scalar wave equation in three‐dimensions

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Abstract
A novel method has been proposed for the finite element solution of the steady‐state scalar wave equation in three‐dimensions. In this the governing equation and the prescribed boundary conditions in the physical space are transformed into a spherical polar space in which the radial direction is logarithmically condensed; the physical problem domain is also mapped into the new space. The transformed equation is then solved in the mapped domain using conventional finite elements. Because physical dimensions of the problem are logarithmically condensed in the proposed spherical polar space, the method is particularly suitable for solving truly three‐dimensional problems in which the aspect ratio(s) is large or very large.A number of illustrative examples considered show that the proposed method is capable of a high degree of accuracy, achieved efficiently and economically. A hybrid scheme has also been proposed for dealing with awkward‐shaped domains.

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