非线性抛物型方程在流体动力学、气象预报和惯性约束核聚变等领域中有广泛应用。对这类非线性问题，设计适用于复杂区域和网格并且具有保正性的格式是重要的研究问题。本项目拟研究这类非线性问题的虚拟元方法及其自适应算法、新型的时间外推算法与虚拟元的外推瀑布多网格方法，从而构造适用于复杂区域和网格、具有保正性的高精度离散格式，建立相应的数学理论；将所设计的离散格式推广应用到三温辐射扩散方程组，发展对网格变形稳健的离散格式。同时，通过项目的合作研究，为合作单位培养年青学者。

Nonlinear parabolic equations are widely used in the fluid dynamics, weather forecasting, inertial confinement fusion and other related fields. It is an important topic to design positivity preserving schemes which are able to adopt to both complicated domains and complicated grids, of this kind of nonlinear problems. This proposal is to study virtual element methods, corresponding adaptive algorithms and extrapolation cascadic multigrid methods, time extrapolation algorithms, of this class of nonlinear problems. One aim is to construct positivity preserving high accurate numerical methods which are able to adopt to both complicated domains and complicated grids for them. The other aim is to extend these schemes to three temperature radiation diffusion equations and develop robust approximate methods with respect to mesh deformation for them. Above all, through this collaboration, the young co-investigator will become stronger and stronger.