Removal of spurious outlier frequencies and modes from isogeometric discretizations of second- and fourth-order problems in one, two, and three dimensions

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

“…It turns out that our strategy is very similar (and actually identical in several cases) to the one proposed in [17]. However, our path towards it is more theoretical and allows us to equip the numerical process with a solid mathematical foundation.…”

“…More recently, interesting contributions have been presented in [9,17], where they exploit the imposition of suitable additional boundary conditions in a similar manner to the optimal spline spaces from [12,14,26]. In [9] a penalization of high-order derivatives near the boundary is proposed to remove the outliers from the isogeometric approximation of the spectrum of the Laplace operator.…”

“…This penalization approach deeply mitigates the spurious frequencies. In [17] the same high-order derivatives at the boundary are strongly set equal to zero, by using suitable spline subspaces as trial spaces. The approach is also tested for fourth-order operators.…”

“…Both the approaches in [9,17] rely on the observation that the exact eigenfunctions satisfy additional homogeneous boundary conditions and on the intuition that adding such features of the exact solution in the discretization (in weak or strong form) would help in fixing the outlier issue. However, a theoretical foundation of the proposed procedures and a proper analysis of the approximation properties of the used spline subspaces and so of the accuracy of the whole process are missing.…”

“…The spline subspaces used in [17] as trial spaces are not new in the literature. They are the "reduced spline spaces" considered in [27, Section 5.2] (see also [26,34] for some special cases).…”