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.

翻译：模拟装置内嵌材料的微结构进化往往涉及整体边界条件。 在这里,我们提出修改的尼采用一个整体边界条件解决普瓦森方程式的方法,这与在金属-内聚体转换过程中的紧密关联材料的微结构进化的阶段-实地方程相结合。我们的数字实验表明,拟议的方法实现了最佳趋同率,而常规拉格朗乘数法的趋同率则不是最佳的。此外,由修改的尼采方法产生的线性系统可以通过一个具有代数多电网先决条件的迭代求解器来解决。修改的尼采方法可以在数学上类似于这种电动整体边界条件的其他物理边界条件中应用。