A reduced-order LSMFE formulation based on POD method and implementation of algorithm for parabolic equations

Abstract

Proper orthogonal decomposition (POD) technology has been successfully used in the reduced-order modeling of complex systems. In this paper, we extend the applications of POD method to a usual least-squares mixed finite element (LSMFE) formulation for two-dimensional parabolic equations with real practical applied background. POD bases here are constructed using the method of snapshots. Karhunen–Loeve projection is used to develop a reduced-order LSMFE formulation obtained by projecting the usual LSMFE formulation onto the most important POD bases. The reduced-order LSMFE formulation has lower dimensions and sufficiently high accuracy, is suitable for finding approximate solutions for two-dimensional parabolic equations, and can circumvent the constraint of the convergence stability what is called Brezzi–Babuška condition. We derive the error estimates between the reduced-order LSMFE solutions and the usual LSMFE solutions for parabolic equations and provide the implementation of algorithm for solving the reduced-order LSMFE formulation so as to supply scientific theoretic basis and computing way for practical applications. Two numerical examples are used to validate that the results of numerical computations are consistent with theoretical conclusions. Moreover, it is shown that the reduced-order LSMFE formulation based on POD method is feasible and efficient for finding numerical solutions of parabolic equations.