On the $h$-adaptive PUM and the $hp$-adaptive FEM approaches applied to PDEs in quantum mechanics

TL;DR

This paper compares $h$-adaptive partition-of-unity and $hp$-adaptive finite element methods for solving PDEs in quantum mechanics, demonstrating high accuracy and efficiency through numerical experiments.

Contribution

It introduces an a posteriori error estimator for the partition-of-unity method and compares its performance with $h$- and $hp$-adaptive FEM in quantum PDEs.

Findings

Partition-of-unity method achieves comparable accuracy to $hp$-FEM.

Error estimators enable effective adaptive mesh refinement.

Numerical results confirm high efficiency and accuracy.

Abstract

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 Schr\"odinger 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 $10^{-3}$.