Numerical Solution for Schrödinger Eigenvalue Problem Using Isogeometric Analysis on Implicit Domains

Abstract

We study the accuracy and performance of isogeometric analysis on implicit domains when solving time-independent Schrodinger equation. We construct weighted extended PHT-spline basis functions for analysis, and the domain is presented with same basis functions in implicit form excluding the need for a parameterization step. Moreover, an adaptive refinement process is formulated and discussed with details. The constructed basis functions with cubic polynomials and only $$C^{1}$$ continuity are enough to produce a higher continuous field approximation while maintaining the computational cost for the matrices as low as possible. A numerical implementation for the adaptive method is performed on Schrodinger eigenvalue problem with double-well potential using 3 examples on different implicit domains. The convergence and performance results demonstrate the efficiency and accuracy of the approach.