The simplified weak Galerkin method with θ scheme and its reduced-order model for the elastodynamic problem on polygonal mesh

This paper presents a simplified weak Galerkin (SWG) method for solving the elastodynamic problem and its reduced-order model (ROM) using the proper orthogonal decomposition (POD) technique. The SWG method allows for the use of polygonal meshes. It only utilizes degrees of freedom associated with the boundary, reducing computational complexity compared to the classical weak Galerkin method. Moreover, we apply the POD technique to develop a POD-SWG-ROM for the problem, further enhancing the computational efficiency. Then, to discretize in time, we utilize a θ-scheme, where the scheme is explicit when $0\le \theta <1/4$ and implicit when $1/4\le \theta \le 1/2$. We establish the theoretical analysis of the semi-discrete scheme and the fully-discrete θ scheme. The theoretical analysis demonstrates that the method is locking-free, and the convergence rate in the $H^1$ and $L^2$ norms is $O(\Delta t^2+h^1)$ and $O(\Delta t^2+h^2)$ respectively. Finally, we verify the theoretical analysis through numerical tests and effectively simulate the propagation of elastic waves under polygonal meshes. Moreover, the proposed POD-SWG-ROM can significantly improve computational efficiency.