The POD-based reduced-dimension study on the two-grid finite element method for the nonlinear time-fractional wave equation

The main purpose of this paper was to study the reduced-dimension of unknown classical two-grid finite element (CTGFE) solution coefficient vectors for the nonlinear time-fractional wave (NTFW) equation by using proper orthogonal decomposition (POD). For this purpose, a CTGFE method with unconditional stability for the NTFW equation and the error estimates of CTGFE solutions were reviewed. Then, the CTGFE method was rewritten into matrix form, and the unknown solution coefficient vectors in the matrix CTGFE method were reduced by the POD method, so a new reduced-dimension TGFE (RDTGFE) method was created. The biggest contribution of this paper consists in analyzing theoretically the existence, stability, and errors of the RDTGFE solutions, and in applications, verifying the correctness of the obtained theoretical results and the advantages of the RDTGFE method. The RDTGFE method can not only greatly reduce the unknowns of the CTGFE method and the simplify computational process but also greatly save CPU runtime and improve the computational efficiency. Therefore, the RDTGFE method is worth studying and spreading.