Space-time least squares approximation for Schrödinger equation and efficient solver
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The application of Galerkin methods to time-dependent phenomena has traditionally relied on step-by-step time-marching schemes, typically using either finite difference methods or discontinuous Galerkin methods in time combined with continuous Galerkin methods in space. More recently, the concept of coherent discretization over the entire space-time domain has gained attention as a means to better leverage modern parallel computational resources. In this context, the authors propose a spline based space-time least squares discretization of the Schrödinger equation on Cartesian domains. Alongside this, exploiting the tensor product structure of the basis functions, the authors introduce a preconditioner that is written as the sum of Kronecker products of matrices and is applied using an alternative version of the classical Fast Diagonalization (FD) method. This ensures efficiency and robustness, regardless of the polynomial degree of the spline space. Notably, the application time scales almost linearly with the number of degrees of freedom in serial computations.