Iterative solution of Helmholtz problem with high-order isogeometric analysis and finite element method at mid-range frequencies

Abstract

Solving wave problems with isogeometric analysis has attracted a significant attention in the past few years. It is well known that keeping a fixed number of degrees of freedom per wavelength leads to an increased error as higher wavenumbers are considered. This behaviour often cited as the pollution error, improves significantly with isogeometric analysis when compared to the conventional finite element method. The improvement in handling pollution along with the ability to represent exact geometries has been the main reasons behind the attention that isogeometric analysis has received. Furthermore, using high order elements also presents major advantages over low order elements for this range of frequencies. However, it remains to be studied how iterative linear solvers, often necessary for solving high frequency wave problems, perform when using isogeometric analysis compared to the finite element method especially at high polynomial orders. This paper is one of the first studies in this direction.In this work we investigate the Generalised Minimal Residual method, a standard Krylov subspace iterative technique, for solving the linear system resulting form isogeometric analysis. Furthermore, we look into the use of some recently proposed preconditioners for Helmholtz problem, such as shifted Laplace or ILU with a complex shift preconditioners and how they perform with high order isogeometric analysis and finite element method. In general the results show improvement when using isogeometric analysis in terms of the number of iterations required for convergence compared to the finite element method for both preconditioned and non-preconditioned linear systems. We use eigenvalue spectra to understand this improvement.