2016/12/07 by Denis Davydov, Tymofiy Gerasimov, Davydov, Denis +6
Engineering · Mathematics · Physics and Astronomy · #Advanced Numerical Methods in Computational Mathematics #Computational Physics (physics.comp-ph) #Electromagnetic Simulation and Numerical Methods #FOS: Physical sciences #Numerical methods for differential equations #Numerical methods in engineering #physics.comp-ph
paper · pdf · doi:10.48550/arxiv.1612.02305
arxiv created 2016/12/07 · openalex publication_date 2016/12/07 · arxiv updated 2016/12/08 · openalex created_date 2022/09/02 · openalex updated_date 2026/07/28
In this paper the h-adaptive partition-of-unity method and the h- and hp-adaptive finite element method are applied to partial differential equations arising in quantum mechanics, namely, the Schrödinger equation with Coulomb and harmonic potentials, and the Poisson problem. Implementational details of the partition-of-unity method related to enforcing continuity with hanging nodes and the degeneracy of the basis are discussed. The partition-of-unity method is equipped with an a posteriori error estimator, thus enabling implementation of error-controlled adaptive mesh refinement strategies. To that end, local interpolation error estimates are derived for the partition-of-unity method enriched with a class of exponential functions. The results are the same as for the finite element method and thereby admit the usage of standard residual error indicators. The efficiency of the h-adaptive partition-of-unity method is compared to the h- and hp-adaptive finite element method. The latter is implemented by adopting the analyticity estimate from Legendre coefficients. An extension of this approach to multiple solution vectors is proposed. Numerical results confirm the remarkable accuracy of the h-adaptive partition-of-unity approach. In case of the Hydrogen atom, the h-adaptive linear partition-of-unity method was found to be comparable to the hp-adaptive finite element method for the target eigenvalue accuracy of 10-3.