Discretization requirements: How many elements per wavelength are necessary?

The commonly applied rule of thumb to use a fixed number of elements per wavelength in linear time-harmonic acoustics is discussed together with the question of using either continuous or discontinuous elements for collocation. Continuous interpolation of the sound pressure has been favored in most applications of boundary element methods for acoustics. Only a few papers are known where discontinuous elements are applied because they guarantee C1 continuity of the geometry at element edges. In these cases, it was assumed that the same number of elements as for continuous elements is required for the same numeric error. Of course this implies a larger degree of freedom. An effect of superconvergence is known for boundary element collocation on discontinuous elements. This effect is observed if the collocation points are located at the zeros of orthogonal functions, e.g. at the zeros of the Legendre polynomials. We start with a review of continuous and discontinuous boundary elements using constant, linear and quadratic interpolation on triangular and quadrilateral elements. Major part of this contribution consists of the investigation of the computational example of a long duct. For that, the numeric solution is compared with the analytic solution of the corresponding one-dimensional problem. Error dependence in terms of frequency, element size and location of nodes on discontinuous elements is reported. It will be shown that the zeros of the Legendre polynomials account for an optimal position of nodes for this problem of interior acoustics. Similar results are observed for triangular elements. It can be seen that the error in the Euclidean norm changes by one or two orders of magnitude if the location of nodes is shifted over the element. The irregular mesh of a sedan cabin compartment accounts for the second example. The optimal choice of node position is confirmed for this example. It is one of the key results of this paper, that discontinuous boundary elements perform more efficiently than continuous ones, in particular for linear elements. This, however, implies that nodes are located at the zeros of orthogonal functions on the element. Furthermore, there is no indication of a similarity to the pollution effect which is known from finite element methods.