Higher order finite and infinite elements for the solution of Helmholtz problems

In recent years the focus in the field of acoustic finite element computations has shifted to the mid-frequency regime. At higher frequencies, however, the conventional finite elements with linear shape functions fail to provide reliable results due to so-called pollution effects. This motivates the use of higher order shape functions and related p-FEM concepts. The solution of large systems of equations arising from engineering problems often involves the use of iterative solution procedures, mostly by means of Krylov-subspace methods. The performance of these methods strongly depends on the spectrum of the resulting system matrices, which is affected by the polynomial approximation within the finite element formulation.The current work shows that finite elements based on Bernstein polynomials yield a favorable spectrum of the system matrix and particularly good performance in combination with commonly employed Krylov solvers. This is shown for interior as well as exterior Helmholtz problems, where an infinite element formulation is employed to account for the sound radiation.