Monica Montardini

Monica Montardini, Giancarlo Sangalli, Mattia Tani

In this paper we propose a new class of preconditioners for the isogeometric discretization of the Stokes system. Their application involves the solution of a Sylvester-like equation, which can be done efficiently thanks to the Fast Diagonalization method. These preconditioners are robust with respect to both the spline degree and mesh size. By incorporating information on the geometry parametrization and equation coefficients, we maintain efficiency on non-trivial computational domains and for variable kinematic viscosity. In our numerical tests we compare to a standard approach, showing that the overall iterative solver based on our preconditioners is significantly faster.

Monica Montardini, Filippo Remonato, Giancarlo Sangalli

We present in detail three different quasi-Newton isogeometric algorithms for the treatment of free boundary problems. Two algorithms are based on standard Galerkin formulations, while the third is a fully-collocated scheme. With respect to standard approaches, isogeometric analysis enables the accurate description of curved geometries, and is thus particularly suitable for free boundary numerical simulation. We apply the algorithms and compare their performances to several benchmark tests, considering both Dirichlet and periodic boundary conditions. In this context, iogeometric collocation turns out to be robust and computationally more efficient than Galerkin. Our results constitute a starting point of an in-depth analysis of the Euler equations for incompressible fluids.

Monica Montardini, Giancarlo Sangalli, Lorenzo Tamellini

In this paper we investigate numerically the order of convergence of an isogeometric collocation method that builds upon the least-squares collocation method presented in [1] and the variational collocation method presented in [2]. The focus is on smoothest B-splines/NURBS approximations, i.e, having global $C^{p-1}$ continuity for polynomial degree $p$. Within the framework of [2], we select as collocation points a subset of those considered in [1], which are related to the Galerkin superconvergence theory. With our choice, that features local symmetry of the collocation stencil, we improve the convergence behaviour with respect to [2], achieving optimal $L^2$-convergence for odd degree B-splines/NURBS approximations. The same optimal order of convergence is seen in [1], where, however a least-squares formulation is adopted. Further careful study is needed, since the robustness of the method and its mathematical foundation are still unclear.