本项目拟根据相场数学模型及相关数学问题本身的特性，将网格剖分、加密和粗化、移动，时间和空间离散方法，非线性迭代方法和非线性多重网格方法等作为一个有机整体来研究，发展这类问题精度高、计算复杂度少，且具有数学理论支持的新型数值方法。这包括研究高维高阶非线性奇异摄动问题新型自适应有限元方法，设计高效、鲁棒非线性迭代方法和非线性多重网格方法，发展大步长时间自适应方法，研究非线性迭代方法的初值选取，并将这些高效算法和高质量网格生成快速算法与超收敛有机结合，获得非稳态参数相关强非线性问题高精度算法。

In this project, all factors of numerical methods, including mesh generation, refining, coarsening and moving, discretizations for both spatial and temporal variables, nonlinear iterative methods and nonlinear multigrid methods, will be studied in an integrated fashion for mathematical models of phase fields and related mathematical problems, where corresponding properties of these problems will be fully explored. The final aim is to develop new numerical methods with both high accuracy and less complexity for this class of problems, and establish their mathematical theories. In particular, this will lead to new adaptive finite element methods for nonlinear singularly perturbed problems of high order in high dimension, high accurate and robust nonlinear iterative and multigrid methods, adaptive time stepping methods with large steps, properly chosen initial guesses for nonlinear iterative methods. Finally, these high accurate methods will be combined with fast algorithms for meshes generation of high quality and superconvergence to design numerical methods of high accuracy for time and parameter dependent nonlinear problems.