An efficient finite element method applied to quantum billiard systems

TL;DR

This paper introduces an efficient finite element method using $C_1$ continuity quartic interpolation for calculating eigenvalues and eigenfunctions in quantum billiard systems, achieving high accuracy across various shapes.

Contribution

The paper presents a novel FEM approach with $C_1$ continuity quartic interpolation that outperforms traditional $C_0$ quadratic FEM in quantum billiard calculations.

Findings

Accurately computes over a thousand eigenvalues for different billiard shapes.

Demonstrates superior efficiency compared to $C_0$ quadratic FEM.

Applicable to complex quantum billiard geometries.

Abstract

An efficient finite element method (FEM) for calculating eigenvalues and eigenfunctions of quantum billiard systems is presented. We consider the FEM based on triangular $C_1$ continuity quartic interpolation. Various shapes of quantum billiards including an integrable unit circle are treated. The numerical results show that the applied method provides accurate set of eigenvalues exceeding a thousand levels for any shape of quantum billiards on a personal computer. Comparison with the results from the FEM based on well-known $C_0$ continuity quadratic interpolation proves the efficiency of the method.