Outlier-free spline spaces for isogeometric discretizations of biharmonic and polyharmonic eigenvalue problems

Abstract

We present outlier-free isogeometric Galerkin discretizations of eigenvalue problems related to the biharmonic and the polyharmonic operator in the univariate setting. These are Galerkin discretizations in certain spline subspaces that provide accurate approximations of all eigenfrequencies and eigenmodes, without the occurrence of spurious outliers. The theoretical cornerstone is the use of spline subspaces that are optimal in the sense of L2 Kolmogorov n-widths for special function classes related to the eigenvalue problems under investigation. This work is a continuation and extension of a similar study, recently presented for the Laplace operator, towards higher-order problems. As for the Laplace operator, the considered optimal spline spaces are identified by additional homogeneous boundary conditions and by specific sequences of break points. For fourth- and higher-order problems, however, optimal spline spaces are not known for all degrees and the optimal break points are not explicitly given. A careful analysis of the properties of optimal spaces allows us to determine spline subspaces of practical interest that are outlier-free. In particular, in the biharmonic case, we are able to construct such outlier-free spline spaces, for any odd and even degree, by considering break points that are uniform, or just a minor modification of the uniform ones. This improves upon the (numerical) results available in the literature. The theoretical findings are validated by a selection of numerical tests.