Integral boundary conditions in phase field models

TL;DR

This paper introduces a novel variational approach for solving PDEs with integral boundary conditions in phase field models, demonstrating improved convergence and computational efficiency over traditional methods.

Contribution

A new variational formulation for integral boundary conditions in phase field models that outperforms the Lagrange multiplier method in convergence and computational efficiency.

Findings

Achieves optimal convergence rate.

Produces symmetric positive definite linear systems.

Enables efficient solution with Conjugate Gradient and multigrid preconditioning.

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.