Isogeometric Analysis (IgA) is an evolution of the finite element method: it employs B-splines or their generalization both to represent the computational domain and to approximate the solution of the considered partial differential equation. The high-continuity of isogeometric basis functions leads to several advantages, e.g. higher accuracy per degree-of-freedom, but it introduces challenging problems at the computational level: one of the major issues is the efficient solution of linear systems. The development of computationally efficient and robust preconditioners is a challenge in IgA community. Indeed, a good preconditioner helps the convergence of iterative methods, that in IgA are preferred to the direct ones. In this talk, after a brief introduction to IgA, I will focus on the study of an efficient solver for a Galerkin space-time isogeometric discretization of the heat equation. Exploiting the tensor product structure of the basis functions in the parametric domain, I propose a preconditioner that can be efficiently applied thanks to an extension of the classical Fast Diagonalization method. The preconditioner is robust with respect to the polynomial degree and the time required for the application is almost proportional to the number of degrees-of-freedom.
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