研究若干非线性发展方程的tau-Chebyshev配置时空谱元方法：1)常微分方程初值问题的tau-Chebyshev配置谱元方法及其并行算法，在局部Lipschitz条件下获得误差估计，分析超收敛性质；2)偏微分方程边值问题的tau-Chebyshev配置谱元方法，对不同的边值条件，设计tau谱元格式，建立反映特征和耗散性质的能量关系，获得稳定性，应用于KdV、Maxwell和Burgers方程等，获得最优误差估计；3)建立tau-Chebyshev配置时空谱元方法，设计时空高度并行化显隐迭代格式，非线性部分可基于Chebyshev点快速计算，形成具有时空谱精度的高效算法，获得误差估计。结合前期三角单元和四面体单元谱方法，建立较一般区域上发展方程的tau-Chebyshev配置时空谱元方法；4)对具奇性问题和分数阶微分方程建立Jacobi tau-Chebyshev配置谱元方法。

The tau-Chebyshev collocation time-space spectral element methods are to be developed for some nonlinear evolution equations: 1)The tau-Chebyshev collocation spectral element method and its parallel algorithm for initial value problems of ordinary differential equations. Obtain error estimates under the local Lipschitz condition, and analyze its superconvergence properties. 2)Design tau-Chebyshev collocation spectral element methods for problems of partial differential equations with different boundary value conditions, and establish the energy relation reflecting the characteristics and dissipative properties to achieve better stability. Apply the method to the KdV equation, Maxwell equation and Burgers equation, and obtain the optimal error estimate. 3)Develop the tau-Chebyshev collocation time-space spectral element methods, design the highly parallel explicit-implicit iteration methods so that the nonlinear part can be computed efficiently by the Chebyshev collocation to form efficient algorithm of spectral accuracy both in time and space, and obtain their error estimates. Combining the previous works on triangular element and tetrahedron element spectral methods, develop tau-Chebyshev collocation time-space spectral element methods for problems on complex domains. 4)Present the Jacobi tau-Chebyshev collocation spectral element methods for singular problems and fractional differential equations.