A reduced-order extrapolation technique for solution coefficient vectors in the mixed finite element method for the 2D nonlinear Rosenau equation

Abstract

This study focuses mainly on a reduced-order method for the coefficient vectors in the Crank-Nicolson mixed finite element (CNMFE) solutions for the two-dimensional nonlinear Rosenau equation. We first provide a CNMFE method for the Rosenau equation and give relative theoretical results as well as rewrite the CNMFE method in matrix form. We then produce two sets of proper orthogonal decomposition (POD) bases by means of the first few coefficient vectors of the CNMFE solutions and set up a reduced-order extrapolating CNMFE (ROECNMFE) model with the POD bases as well as discuss the existence and stability as well as errors of the ROECNMFE solutions by means of the matrix tool, resulting in the theoretical analysis becoming very succinct. Lastly, we present some numerical experiments to validate the correctness of the theoretical results, finding that the superiority of the ROECNMFE method is further verified.