Chapter 2 - Reduced-Order Extrapolation Finite Element Methods Based on Proper Orthogonal Decomposition

Abstract

In comparison with the FD method in Chapter 1, the finite element (FE) method has a shorter history of development. However, regarding both computation and theory, FE has great(er) advantages. The use of FE of various shapes and isoparametric elements enable one to approximate complex geometries much better. Theoretically, FE also ties in well with calculus of variations and the estimates in Sobolev spaces. In the past two decades, more and more automated, open-access FE software has emerged, facilitating and accelerating FE coding and computer implementation. The technical development of this chapter will follow the same stylistic format as Chapter 1. First, in order to make this chapter sufficiently self-contained, we first give a concise review of the basic theory of the classical FE method and the classical mixed FE (MFE) method. Afterwards, we introduce the construction, theoretical analysis, and implementations of algorithms for the POD-based reduced-order extrapolation FE (PODROEFE) methods for the 2D viscoelastic wave equation and 2D nonstationary Burgers equation as well as the POD-based reduced-order stabilized Crank–Nicolson (CN) extrapolation MFE (PODROSCNEMFE) method for the 2D parabolized Navier–Stokes equation. Finally, we provide numerical examples to show that the PODROEFE methods and the PODROSCNEMFE method are more advantageous than the classical FE methods and the classical MFE method, respectively. Moreover, it is also shown that the PODROEFE methods and PODROSCNEMFE method are reliable and effective for finding the numerical solutions of the 2D viscoelastic wave equation, the 2D nonstationary Burgers equation, and the 2D parabolized Navier–Stokes equation.