Integral boundary conditions in phase field models

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 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.

Automated summary

Modeling the transport and evolution process of a material in an electronic device involves solving partial differential equations with integral boundary conditions. In this study, the authors propose a new method that efficiently handles such boundary conditions and achieves optimal convergence rate compared to the commonly-used Lagrange multiplier method. Numerical experiments also show that the linear system derived from the new method can be efficiently solved by the Conjugate Gradient method with algebraic multigrid preconditioning.