Some Studies on Adaptive Finite Element Method (AFEM) for Schrödinger Equation of Hydrogen Atom
Graduate Research, LSEC, UCAS, 2019
- Advisor: Prof. Aihui Zhou & Xiaoying Dai
- Date: Sep 2017 - May 2019
- Reviewed the basic theories and algorithms of AFEM for elliptic boundary value and eigenvalue problems, including a priori error estimation, a posteriori error estimation, adaptive mesh-refinement techniques, convergence rate and optimal complexity; oral presented at group meetings
- Reviewed the first principles electronic structure calculations, comprising the mathematical foundation and physical background
- Wrote shell scripts under Linux environment
- Parallel Calculated the finite element solutions to nonlinear partial differential equations, using C language based on Parallel Hierarchical Grid (PHG) package
- Visualized the 3D spherical harmonics solutions via ParaView and analyzed the accuracy of the finite element method with different iterations
- Designed algorithms to reduce the computation error and iterations caused by hydrogen atom’s small spectral gap in numerical solutions to generalized linear eigenvalue problems
- Proposed a more proper arrangement of initial mesh, Improved the effective polynomial order for the finite element in the discretization of Kohn-Sham equation
- Designed two integration strategies to calculate the singular integral, Suppressed the error caused by the singularity
- Analyzed the computation error from three aspects: model error, numerical error and approximation error
- Reduced the model error and numerical error, Obtained the variation curves of the approximation error which is the core of finite element approximation with the increase of the polynomial order for the finite element