Isogeometric discretizations of evolutionary equations and fast solvers.

The aim of this thesis is to contribute to the development of recent numerical methods in the context of isogeometric analysis: firstly, energy conservative discretizations for initial-boundary value problems in mixed form, here proposed for the wave equation, and secondly, fast preconditioners for solving linear systems arising from space-time discretizations of evolutionary equations, here proposed for heat and Schrödinger equations. In the first part, we analyze the wave equation in mixed form, with periodic and/or Dirichlet homogeneous boundary conditions, and nonconstant coefficients that depend on the spatial variable. For the discretization, the weak form of the second equation is replaced by a strong form, written in terms of a projection operator. The system of equations is discretized with B-splines forming a De Rham complex along with suitable commutative projectors for the approximation of the second equation. The discrete scheme is energy conservative when discretized in time with a conservative method such as Crank-Nicolson. We propose a convergence analysis of the method to study the dependence with respect to the mesh size $h$, with focus on the consistency error. Numerical results show optimal convergence of the error in energy norm, and a relative error in energy conservation for long-time simulations of the order of machine precision. In the second part, we introduce preconditioning techniques based on fast diagonalization methods for space-time isogeometric discretization of the heat equation. Three formulation are considered: the Galerkin approach, a Galerkin $L^2$ least squares form and a continuous least squares approach. For each formulation, the heat differential operator is written as a sum of terms that are Kronecker products of univariate operators. These are used to speed-up the application of the operator in iterative solvers and to construct a suitable preconditioner. Contrary to the fast diagonalization technique for the Laplace equation, where all univariate operators acting on the same direction can be simultaneously diagonalized, in the case of the heat equation this is not possible. Luckily, this can be done up to an additional term that has low rank, allowing for the utilization of arrow-head like factorization or inversion by Sherman-Morrison formula. The proposed preconditioners work extremely well on the parametric domain and, when the domain is parametrized or when the equation coefficients are not constant, they can be adapted and retain good performance characteristics. Finally, We present a space-time least squares isogeometric discretization of the Schrödinger equation and propose a preconditioner for the arising linear system in the parametric domain. Exploiting the tensor product structure of the basis functions, the preconditioner is written as the sum of Kronecker products of matrices. Thanks to an extension to the Fast Diagonalization method, the application of the preconditioner is efficient and robust w.r.t. the polynomial degree of the spline space. The time required for the application is almost proportional to the number of degrees-of-freedoms, for a serial execution.