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Numerical Analysis of Multi-patch Discontinuous Galerkin Isogeometric Method for Full-potential Electronic Structure Calculations

Xiaoxu Li, Xucheng Meng

TL;DR

This work offers a unified analysis framework for the DG-IGA method applied to a class of elliptic eigenvalue problems and provides a rigorous error analysis of the DG-IGA approximations for linear eigenvalue problems.

Abstract

In this paper, we study the multi-patch discontinuous Galerkin isogeometric (DG-IGA) approximations for full-potential electronic structure calculations. We decompose the physical domain into several subdomains, represent each part of the wavefunction separately using B-spline basis functions, possibly with different degrees, on varying mesh sizes, and then combine them by DG methods. We also provide a rigorous {\em a priori} error analysis of the DG-IGA approximations for linear eigenvalue problems. Furthermore, this work offers a unified analysis framework for the DG-IGA method applied to a class of elliptic eigenvalue problems. Finally, we present several numerical experiments to verify our theoretical results.

Numerical Analysis of Multi-patch Discontinuous Galerkin Isogeometric Method for Full-potential Electronic Structure Calculations

TL;DR

This work offers a unified analysis framework for the DG-IGA method applied to a class of elliptic eigenvalue problems and provides a rigorous error analysis of the DG-IGA approximations for linear eigenvalue problems.

Abstract

In this paper, we study the multi-patch discontinuous Galerkin isogeometric (DG-IGA) approximations for full-potential electronic structure calculations. We decompose the physical domain into several subdomains, represent each part of the wavefunction separately using B-spline basis functions, possibly with different degrees, on varying mesh sizes, and then combine them by DG methods. We also provide a rigorous {\em a priori} error analysis of the DG-IGA approximations for linear eigenvalue problems. Furthermore, this work offers a unified analysis framework for the DG-IGA method applied to a class of elliptic eigenvalue problems. Finally, we present several numerical experiments to verify our theoretical results.

Paper Structure

This paper contains 14 sections, 9 theorems, 69 equations, 14 figures, 2 tables.

Table of Contents

  1. Introduction
  2. B-spline approximations
  3. DG multi-patch isogeometric method
  4. Multi-patch discretization
  5. DG approximations of source problem
  6. DG approximations of eigenvalue problem
  7. Numerical results
  8. Concluding remarks
  9. Proofs
  10. Proof of Lemma \ref{['lem:boundness']}
  11. Proof of Lemma \ref{['lem:coercive']}
  12. Proof of Theorem \ref{['thm:T-approximate']}
  13. Proof of Theorem \ref{['thm:error_L2']}
  14. Proof of Theorem \ref{['thm:eigen-source-bound']}

Key Result

Lemma 2.1

Let $0\leq k\leq s$ and $p=\min \{p_i\}$. If $f\in H^{s}([0,1]^d)$, then there exists a positive constant $C$ such that with $t=\min\{p+1,s\}$.

Figures (14)

  • Figure 4.1: The patch decomposition and initial mesh of computational domain.
  • Figure 4.2: (Example 1) The first four reference eigenfunctions.
  • Figure 4.3: (Example 1) The convergence of the first four eigenvalues.
  • Figure 4.4: (Example 1) The convergence of the first eigenfunction.
  • Figure 4.5: (Example 1) The distribution of $|u_1 - u_1^{\rm DG}|$ on three different meshes with $p=2$.
  • ...and 9 more figures

Theorems & Definitions (18)

  • Lemma 2.1: projection error
  • Lemma 3.1: projection error on physical subdomain
  • Lemma 3.2: trace inequality
  • Lemma 3.3: boundness
  • Lemma 3.4: coerciveness
  • Theorem 3.1: error estimate in DG-norm
  • Theorem 3.2: error estimate in $L^2$-norm
  • Theorem 3.3: convergence of eigenvalue problem
  • Remark 3.1: repeated eigenvalues
  • Theorem 3.4: error estimate of eigenvalue problem
  • ...and 8 more