Title: The Partition of Unity Finite Element Method for the Schrödinger Equation

Journal Article · · Computational Methods in Applied Mathematics
 [1];  [2];  [3];  [4]
  1. Computer, Electrical and Mathematical Science and Engineering Division , King Abdullah University of Science and Technology , Thuwal , Saudi Arabia , and Dipartimento di Matematica “F. Casorati”, Università di Pavia, Pavia, Italy
  2. CCS-3 , 5112 Los Alamos National Laboratory , Los Alamos , New Mexico , USA
  3. Dipartimento di Matematica “F. Casorati” , 19001 Università di Pavia , Pavia , Italy
  4. T-5 , 5112 Los Alamos National Laboratory , Los Alamos , New Mexico , USA
+ Show Author Affiliations

Abstract A Schrödinger equation for the system’s wavefunctions in a parallelepiped unit cell subject to Bloch-periodic boundary conditions must be solved repeatedly in quantum mechanical computations to derive the materials’ properties. Recent studies have demonstrated how enriched finite element type Galerkin methods can substantially lower the number of degrees of freedom necessary to produce accurate solutions with respect to the standard plane-waves method. In particular, the flat-top partition of unity finite element method enriched with the radial eigenfunctions of the one-dimensional Schrödinger equation offers a very effective way of solving the three-dimensional Schrödinger eigenvalue problem. We investigate the theoretical properties of this approximation method, its well-posedness and stability, we prove its convergence and derive suitable bound for the ℎ- and 𝑝-refinement in the L 2 L^{2} and energy norm for both the eigenvalues and the eigenfunctions. Finally, we confirm these theoretical results by applying this method to the eigenvalue problem of the one-electron Schrödinger equation with the harmonic potential, for which the exact solution is known.

Research Organization:
Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
Sponsoring Organization:
USDOE; USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA)
Grant/Contract Number:
89233218CNA000001
OSTI ID:
2350868
Report Number(s):
LA-UR-23-25197
Journal Information:
Computational Methods in Applied Mathematics, Journal Name: Computational Methods in Applied Mathematics; ISSN 1609-4840
Publisher:
Walter de Gruyter GmbHCopyright Statement
Country of Publication:
Germany
Language:
English

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97 MATHEMATICS AND COMPUTING
Enriched Partition of Unity Finite Element Method
Mathematics
Schrödinger Eigenvalue Problem