Application of optimal spline subspaces for the removal of spurious outliers in isogeometric discretizations

TL;DR

This paper demonstrates that specific optimal spline subspaces in isogeometric analysis effectively eliminate spurious outliers in eigenvalue discretizations of the Laplace operator, ensuring accurate spectrum approximation.

Contribution

It provides a detailed analysis and explicit error estimates for these optimal spline subspaces, confirming their effectiveness in outlier-free eigenvalue approximations.

Findings

No outliers in eigenvalue discretizations within optimal spline subspaces.

Explicit $L^2$ and $H^1$ error estimates with full approximation order.

Accurate eigenfunction and eigenvalue approximation with fixed degrees of freedom.

Abstract

We show that isogeometric Galerkin discretizations of eigenvalue problems related to the Laplace operator subject to any standard type of homogeneous boundary conditions have no outliers in certain optimal spline subspaces. Roughly speaking, these optimal subspaces are obtained from the full spline space defined on certain uniform knot sequences by imposing specific additional boundary conditions. The spline subspaces of interest have been introduced in the literature some years ago when proving their optimality with respect to Kolmogorov $n$-widths in $L^2$-norm for some function classes. The eigenfunctions of the Laplacian -- with any standard type of homogeneous boundary conditions -- belong to such classes. Here we complete the analysis of the approximation properties of these optimal spline subspaces. In particular, we provide explicit $L^2$ and $H^1$ error estimates with full approximation order for Ritz projectors in the univariate and in the multivariate tensor-product setting. Besides their intrinsic interest, these estimates imply that, for a fixed number of degrees of freedom, all the eigenfunctions and the corresponding eigenvalues are well approximated, without loss of accuracy in the whole spectrum when compared to the full spline space. Moreover, there are no spurious values in the approximated spectrum. In other words, the considered subspaces provide accurate outlier-free discretizations in the univariate and in the multivariate tensor-product case. This main contribution is complemented by an explicit construction of B-spline-like bases for the considered spline subspaces. The role of such spaces as accurate discretization spaces for addressing general problems with non-homogeneous boundary behavior is discussed as well.