Application of optimal spline subspaces for the removal of spurious outliers in isogeometric discretizations
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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 that preserve full approximation power. 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 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 any frequency. Thus, there are no spurious numerical values in the approximated spectrum. In other words, the considered subspaces provide fully 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. Their role as discretization spaces for addressing general problems with non-homogeneous boundary behavior is discussed as well.
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