非线性Galerkin算法的收敛性和复杂性

CONVERGENCE AND COMPLEXITY OF NONLINEAR GALERKIN ALGORITHMS

本文给出了数值求解非线性发展方程的全离散非线性Ｇａｌｅｒｋｉｎ算法，即将空间离散时的谱非线性Ｇａｌｅｒｋｉｎ算法和时间离散的Ｅｕｌｅｒ差分格式相结合，得到了显式和隐式两种全离散数值格式，相应地也考虑了显式和隐式的Ｇａｌｅｒｋｉｎ全离散格式，并分别分析了上述四种全离散格式的收敛性和复杂性，经过比较得出结论：在某些约束条件下，非线性Ｇａｌｅｒｋｉｎ算法和Ｇａｌｅｒｋｉｎ算法具有相同阶的收敛速度。

The paper provides the fully discrete nonlinear Galerkin algorithms for solving the evolution equations.Here the spatial discretization can be performed by the spectral nonlinear Galekin algorithm;time discretization is done by the Euler explicit and implicit schemes.Also,the fully discrete explicit and implicit Galerkin algorithms are considered. Moreover, the convergence and complexity of the above schemes are analysed.It is found that in some situations,the nonlinear Galerkin algorithms are of the convergence rate of same order as ones of Galerkin algorithms.However,the nonlinear Galerkin algorithms are simpler than the Galerkin algorithms.