The reduced-dimension technique for the unknown solution coefficient vectors in the Crank–Nicolson finite element method for the Sobolev equation

Abstract

In this paper, we mainly deal with the reduced-dimension of coefficient vectors of unknown solutions for the Crank–Nicholson finite element (CNFE) method of the Sobolev equation. For this purpose, we first establish the CNFE methods of the Sobolev equation in both functional form and matrix form, and provide the existence, stability, and error estimates of CNFE solutions. We then build a CNFE reduced-dimension recursive (CNFERDR) method in the matrix-form by using a proper orthogonal decomposition (POD) technique, and discuss the existence, stability, and error estimates of CNFERDR solutions by using matrix analysis. In this case, the CNFERDR method includes only few unknowns, but has the same basis functions and accuracy as the CNFE method. Finally, we use two numerical examples to verify the effectiveness of the CNFERDR method.