This study presents a new framework to solve the coupled semilinear elliptic equation by the domain decomposition algorithm. Unlike the traditional domain decomposition algorithm, the coupled semilinear elliptic equation doesn't need to be solved directly. The strategy is to construct a set of nested finite element spaces, and subsequently solve some decoupled linear elliptic equations by using the domain decomposition method in each level space. Additionally, a small-scale coupled semilinear elliptic equation in a specially designed correction space will be solved. As the large-scale coupled semilinear elliptic equation doesn't need to be solved directly, there will be an improved efficiency as compared to the traditional domain decomposition method. Furthermore, as the domain decomposition method is only used to solve decoupled linear elliptic equations, any efficient algorithms designed for the associated linear elliptic equations can be incorporated in the proposed algorithm framework. Thus, the algorithm is highly flexible. Additionally, it can be theoretically proven that the proposed algorithm has very low requirements for the smoothness of nonlinear terms.

中文翻译：

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耦合半线性椭圆方程的域分解新方法

本研究提出了一种新的框架，通过域分解算法求解耦合半线性椭圆方程。与传统的域分解算法不同，耦合半线性椭圆方程不需要直接求解。策略是构造一组嵌套的有限元空间，然后在每一层空间中使用域分解方法求解一些解耦的线性椭圆方程。此外，还将在专门设计的校正空间中求解小尺度耦合半线性椭圆方程。由于大尺度耦合半线性椭圆方程不需要直接求解，与传统的域分解方法相比，效率会有所提高。此外，由于域分解方法仅用于求解解耦线性椭圆方程，因此可以将任何针对相关线性椭圆方程设计的有效算法纳入所提出的算法框架中。因此，该算法是高度灵活的。此外，可以从理论上证明，所提出的算法对非线性项的平滑性要求非常低。