Espen Sande: Outlier-free isogeometric discretizations

In this talk we discuss recent techniques to obtain a priori error estimates with explicit constants for the L2-projection and Ritz-type projections onto spline spaces of arbitrary smoothness defined on arbitrary grids. The presented error estimates indicate that smoother spline spaces exhibit better approximation per degree of freedom, even for low regularity of the function to be approximated. This is in complete agreement with the numerical evidence found in the literature. We further discuss how our results can be used to show that isogeometric discretizations of eigenvalue problems for the Laplacian, with any standard type of homogeneous boundary conditions, have no spurious outlier modes when using certain optimal spline subspaces. Our estimates imply that, for a fixed number of degrees of freedom, all eigenfunctions and eigenvalues of the Laplacian are well approximated, without loss of accuracy in the higher modes.