Nonlinear space-time isogeometric analysis with matrix-free and fast diagonalization methods
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IsoGeometric Analysis, introduced by Hughes et al., is an ``extension'' of Finite Element Methods which obtains better results by increasing the polynomial degree p and refining the mesh size h. However, its application implies a great computational challenge since the cost increases rapidly with the degree p. This communication focuses on novel techniques like Matrix-Free \cite{MF} and Fast Diagonalization \cite{FD} methods which, coupled with IGA, enable to: -reduce storage memory and computation time by not constructing/storing the matrices but instead using an iterative solver with efficient matrix-vector products; -enhance the convergence rate of the Krylov solver by introducing an inexpensive, easy to code and effective preconditioner. The objective of this communication is to illustrate the synergy between these approaches, applied to space-time heat transfer problems, to allow significant cost reductions. In addition, we consider a material nonlinearity case, and we explore the different aspects of the Newton-Krylov iterative solver: stopping criteria for the outer and inner loops, performance of the preconditioner for the inner solver, the benefits of different approximations of the tangent matrix, etc.
