A novel domain decomposition method for coupled semilinear elliptic equation

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.

Automated summary

This study introduces a new framework for solving coupled semilinear elliptic equations by constructing a set of nested finite element spaces and using domain decomposition method to solve decoupled linear elliptic equations in each level space. The proposed algorithm is highly flexible and has very low requirements for the smoothness of nonlinear terms, which improves efficiency compared to traditional domain decomposition methods.