Two‐Grid Model Dimension Reduction Technique for Time‐Fractional Order and Spacial Fourth‐Order Sine‐Gordon Equation

ABSTRACT This article focuses mainly on the dimensionality reduction for the two‐grid Crank‐Nicolson (CN) mixed finite element (MFE) (TGCNMFE) model for the temporal fractional‐order and spatial fourth‐order derivatives sine‐Gordon (TFOSFDOSG) equation with actual applied background. To do so, we first split the TFOSFDOSG equation into two nonlinear equations with second‐order derivatives by introducing an auxiliary function, and construct the TGCNMFE model through using the CN technique discretizing the time derivatives, and the TGCNMFE method discretizing the space variables. The TGCNMFE model is made up of a nonlinear system on coarser meshes together with a linear system on adequately fine meshes, and can be easy to calculate. Thereafter, most importantly, we create a novel TGCNMFE reduced‐dimension (TGCNMFERD) model by means of proper orthogonal decomposition, lowering the dimension of the unknown coefficient vectors of the TGCNMFE solutions, keeping the TGMFE basis functions unchanged, namely keeping the precision unvaried. The main contribution of this article is to theoretically establish the existence, stability, and error estimates of the TGCNMFERD solutions, and in simulation, to confirm the rightness for the obtained theoretical results and the benefit of the TGCNMFERD model through two numerical examples.