Abstract
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.
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