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.

我们表明，与受任何标准类型齐次边界条件约束的拉普拉斯算子相关的特征值问题的等几何伽辽金离散化在某些最优样条子空间中没有异常值。粗略地说，这些最优子空间是通过施加特定的附加边界条件从在某些统一节点序列上定义的完整样条空间获得的。几年前，当证明它们关于 Kolmogorov 的最优性时，文献中已经引入了感兴趣的样条子空间$n$-宽度在 ${升}^2$- 一些函数类的规范。拉普拉斯算子的本征函数——具有任何标准类型的齐次边界条件——都属于此类。这里我们完成了对这些最优样条子空间的逼近性质的分析。特别是，我们提供明确的${升}^2$ 和 $H^1$ 单变量中 Ritz 投影仪的完全近似阶的误差估计并在多元张量积设置中。除了它们的内在兴趣之外，这些估计意味着，对于固定数量的自由度，所有特征函数和相应的特征值都很好地近似，与完整样条空间相比，在整个谱中不会损失精度。此外，在近似频谱中没有杂散值。换句话说，所考虑的子空间在单变量和多变量张量积情况下提供了准确的无异常值离散化。这一主要贡献得到了为所考虑的样条子空间显式构建类 B 样条基的补充。还讨论了此类空间作为用于解决具有非均匀边界行为的一般问题的精确离散化空间的作用。