Space-time least-squares finite elements for evolution equations
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We present first-order least-squares formulations in space-time for the heat and wave equation, and show that these formulations are boundedly invertible, cf [1, 2]. As a consequence, on the one hand, we can define finite element methods which are robust on locally refined space-time meshes, easy to implement, and amenable to conjugate gradient solvers and space-time adaptivity. On the other hand, our formulations are convenient for physics-informed neural networks in space-time, cf. [3]. REFERENCES [1] T. Führer, R. González, and M. Karkulik (2023). Well-posedness of first-order acoustic wave equations and space-time finite element approximation, arXiv. [2] T. Führer and M. Karkulik (2021). Space-time least-squares finite elements for parabolic equations. Comput. Math. Appl., volume 92. [3] J. A. A. Opschoor, P. C. Petersen, C. Schwab (2024). First Order System Least Squares Neural Networks, arXiv.