A novel dimension reduction model based on POD and two-grid Crank-Nicolson mixed finite element methods for 3D nonlinear elastodynamic sine-Gordon problem

Earthquake, petroleum and gas extraction, and underground nuclear test all could cause nonlinear tectonic deformation. It is of great significance to effectively predict and evaluate the disasters and impacts of these geological structure deformation. In this end, a three- dimension (3D) nonlinear elastodynamic sine-Gordon model that can be used to describe nonlinear tectonic deformation including 3D displacement vector, 3 x 3 symmetric stress tensor matrix, nonlinear sin term, and singular initial value functions is first proposed. Then, a new time semi-discrete mixed Crank-Nicolson (TSDMCN) scheme for the 3D nonlinear elastodynamic sine-Gordon model is developed, and the existence, stability, and error estimates of the TSDMCN solutions are proved. Next, a new two-grid mixed finite element Crank-Nicolson (TGMFECN) method with unconditional stability for the 3D nonlinear elastodynamic sine- Gordon model is developed, and the existence, stability, and error estimates of the TGMFECN solutions are proved. Thenceforth, it is the most important thing is that a novel reduced- dimensional iterative TGMFECN (RDITGMFECN) method in matrix form is established by resorting to proper orthogonal decomposition only to lower the unknown TGMFECN solution coefficient vectors and keep TGMFECN basis functions unchanged, which can ensure that the RDITGMFECN method has the same accuracy as the usual TGMFECN method, but can greatly lower the dimension of the unknown TGMFECN solution coefficient vectors so as to mitigate calculated workload, save CPU operating-time, improve computing efficiency, and improve real-time calculating accuracy. In theory, the existence, stability, and error estimates of RDITGMFECN solutions are demonstrated by matrix analysis such that the theoretical analysis becomes very intuitive and easy to be understood by the public, which is a new attempt of theoretical analysis. In application, two numerical examples are used to simulate the 3D nonlinear tectonic deformation caused by earthquake and to verify the correctness of our theoretical results and the effectiveness of the RDITGMFECN method.