Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Integral boundary conditions in phase field models

Xiaofeng Xu, Lian Zhang, Yin Shi, Long-Qing Chen, Jinchao Xu

TL;DR

A modified Nitsche’s method to solve the Poisson equation with an integral boundary condition, which is coupled to phase-field equations of the microstructure evolution of a strongly correlated material undergoing metalinsulator transitions, achieves optimal convergence rate while the rate of convergence of the conventional Lagrange multiplier method is not optimal.

Abstract

Modeling the chemical, electric, and thermal transport as well as phase transitions and the accompanying mesoscale microstructure evolution within a material in an electronic device setting involves the solution of partial differential equations often with integral boundary conditions. Employing the familiar Poisson equation describing the electric potential evolution in a material exhibiting insulator-to-metal transitions, we exploit a special property of such an integral boundary condition, and we properly formulate the variational problem and establish its well-posedness. We then compare our method with the commonly-used Lagrange multiplier method that can also handle such boundary conditions. Numerical experiments demonstrate that our new method achieves an optimal convergence rate in contrast to the conventional Lagrange multiplier method. Furthermore, the linear system derived from our method is symmetric positive definite, and can be efficiently solved by Conjugate Gradient method with algebraic multigrid preconditioning.

Integral boundary conditions in phase field models

TL;DR

A modified Nitsche’s method to solve the Poisson equation with an integral boundary condition, which is coupled to phase-field equations of the microstructure evolution of a strongly correlated material undergoing metalinsulator transitions, achieves optimal convergence rate while the rate of convergence of the conventional Lagrange multiplier method is not optimal.

Abstract

Modeling the chemical, electric, and thermal transport as well as phase transitions and the accompanying mesoscale microstructure evolution within a material in an electronic device setting involves the solution of partial differential equations often with integral boundary conditions. Employing the familiar Poisson equation describing the electric potential evolution in a material exhibiting insulator-to-metal transitions, we exploit a special property of such an integral boundary condition, and we properly formulate the variational problem and establish its well-posedness. We then compare our method with the commonly-used Lagrange multiplier method that can also handle such boundary conditions. Numerical experiments demonstrate that our new method achieves an optimal convergence rate in contrast to the conventional Lagrange multiplier method. Furthermore, the linear system derived from our method is symmetric positive definite, and can be efficiently solved by Conjugate Gradient method with algebraic multigrid preconditioning.

Paper Structure

This paper contains 11 sections, 3 theorems, 38 equations, 1 figure, 6 tables.

Table of Contents

  1. Introduction
  2. Methods
  3. Lagrange multiplier method
  4. New method
  5. Theoretical analysis
  6. Numerical tests
  7. Conclusion
  8. Acknowledgement
  9. Implementation of the Lagrange multiplier method in FEniCS
  10. Basis functions for $V_{h}^c$
  11. Data Availability

Key Result

Theorem 1

(Well-posedness) Assuming that $f \in L^2(\Omega)$, the variational problem (original weak form 2) for the original problem simple model is well-posed if $\sigma$ is a positive function in $\bar{\Omega}$.

Figures (1)

  • Figure 1: A simple model

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof