Comparison of hierarchical basis functions for efficient multilevel solvers

V-cycle and non-standard W-cycle versions of a multiplicative Schwarz preconditioner based on hierarchical basis functions are presented. It is used together with a Krylov subspace method to efficiently solve the linear system that results from higher-order finite element (FE) discretisations of time-harmonic Maxwell equations. A recently developed hierarchical basis for H(curl)-conforming (vector-valued and tangentially continuous) FE spaces on tetrahedral meshes is also briefly presented. On this basis, a certain amount of orthogonality between basis functions of different orders is obtained through the requirement that the Nedelec interpolation of higher-order basis functions vanishes in lower-order FE spaces. Numerical experiments are used to show the good performance of the presented schemes. In these experiments, the performance obtained with the presented basis is compared with the performance obtained by several other hierarchical bases found in the literature. For third-order elements, it is observed that most bases give very similar performance and that the V-cycle preconditioner typically requires about 30% more computing time than the W-cycle one. For fourth-order elements, the new basis combined with the non-standard W-cycle preconditioner leads to the best performance. The computing times for the other combinations are about 40% longer, at best.