Integral boundary conditions in phase field models

Modeling the microstructure evolution of a material embedded in a device often involves integral boundary conditions. Here we propose 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 metal-insulator transitions. Our numerical experiments demonstrate that the proposed method achieves optimal convergence rate while the rate of convergence of the conventional Lagrange multiplier method is not optimal. Furthermore, the linear system derived from the modified Nitsche's method can be solved by an iterative solver with algebraic multigrid preconditioning. The modified Nitsche's method can be applied to other physical boundary conditions mathematically similar to this electric integral boundary condition.