偏微分方程的算子自定义小波解耦算法研究

Operator Custom-Design Decoupling Algorithm for Partial Differential Equations

利用小波算法求解偏微分方程最困难的问题是随着尺度的升高,系统方程的耦合度越来越高,极大降低了计算效率和精度。针对此问题提出了采用算子自定义小波的多尺度解耦算法,首先建立有限元多分辨空间和小波细化关系,提出偏微分方程的多尺度计算理论方法。在优化方案的基础上,提出算子自定义小波的构造方法及解耦条件。改进方法的突出优点在于根据工程问题的实际需要灵活构造具有期望特性的小波基。提出偏微分方程的多尺度算子自定义小波算法,充分利用算子自定义小波的嵌套逼近和尺度解耦特性,实现问题的高效求解。仿真结果表明,改进的算子自定义小波解耦算法具有计算效率高、精度高等特点。

The most difficult problem of solving partial differential equations （PDEs） is that.the decoupling degrees of discrete equations become higher and higher while the scale is increasing, which results in the low computational efficiency and precision. An operator custom - design decoupling algorithm was proposed to solve this problem. The muhiresolution finite element space and wavelet refinement relation were constructed. The operator custom - design wavelets and decoupling condition were proposed based on the lifting scheme. The distinguished feature of the construction method is that the wavelets can be designed depending on the requirements of engineering problems. A muhiscale operator custom - design wavelet decoupling algorithm was presented for solving PDEs, which uses both the nested approximation and scale deeoupling feature. Numerical example demonstrates that the operator custom - design decoupling algorithm is highly efficient and accurate.