The efficient S-DDM scheme and its analysis for solving parabolic equations

In this paper, we develop and analyze the efficient splitting domain decomposition method for solving parabolic equations. The domain is divided into non-overlapping multi-block subdomains. On interfaces of sub-domains, the local multilevel explicit scheme is proposed to solve the interface values of solution, and then the splitting implicit scheme is proposed to solve the solution in interiors of subdomains. On the theoretical aspect, analyzing the S-DDM scheme over non-overlapping multi-block sub-domains is much challenging. We strictly prove the stability and convergence of the S-DDM scheme and obtain the optimal error estimates over non-overlapping multi-block subdomains. Numerical experiments are given to illustrate the accuracy and efficiency of the S-DDM scheme. The developed method takes the most attractive advantages of both the non-overlapping domain decomposition and the splitting technique, which overcomes the limitation of the stripe-divided sub-domains for parabolic equations in the previous methods. The developed S-DDM scheme reduces computational complexities, large memory requirements and computational costs.