Abstract
We present a space–time ultra-weak discontinuous Galerkin discretization of the linear Schrödinger equation with variable potential. The proposed method is well-posed and quasi-optimal in mesh-dependent norms for very general discrete spaces. Optimal h-convergence error estimates are derived for the method when test and trial spaces are chosen either as piecewise polynomials or as a novel quasi-Trefftz polynomial space. The latter allows for a substantial reduction of the number of degrees of freedom and admits piecewise-smooth potentials. Several numerical experiments validate the accuracy and advantages of the proposed method.
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We thank the anonymous reviewers for their valuable comments and suggestions.
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Open access funding provided by Università degli Studi di Milano - Bicocca within the CRUI-CARE Agreement. The authors received support from GNCS-INDAM, from PRIN projects “NA-FROM-PDEs” and “ASTICE”, and from PNRR-M4C2-I1.4-NC-HPC-Spoke6 funded by European Union – NextGenerationEU.
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Gómez, S., Moiola, A. A space–time DG method for the Schrödinger equation with variable potential. Adv Comput Math 50, 15 (2024). https://doi.org/10.1007/s10444-024-10108-9
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DOI: https://doi.org/10.1007/s10444-024-10108-9
Keywords
- Schrödinger equation
- Ultra-weak formulation
- Discontinuous Galerkin method
- Smooth potential
- Quasi-Trefftz space