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：在器件中嵌入材料的微观结构演化建模通常涉及积分边界条件。本文提出一种改进的Nitsche方法，用于求解带有积分边界条件的泊松方程，该方法与描述经历金属-绝缘体相变的强关联材料微观结构演化的相场方程耦合。数值实验表明，所提方法实现了最优收敛速率，而传统拉格朗日乘子法的收敛速率并非最优。此外，由改进的Nitsche方法导出的线性系统可通过代数多重网格预处理的迭代求解器求解。改进的Nitsche方法可推广至与该电积分边界条件数学形式上相似的其他物理边界条件。