A Reduced Weak Galerkin Finite Element Formulation Based on the POD Method for a Semilinear Parabolic Equation

Abstract In this paper, we study a reduced-order model based on proper orthogonal decomposition (POD) for a two-dimensional semilinear parabolic equation. The weak Galerkin finite element method (WG-FEM) is used for spatial discretization and the Crank–Nicolson (CN) scheme is employed for temporal discretization. We introduce the POD formulation and establish the optimal error estimates for both WG-FEM and WG-POD solutions, achieving second-order accuracy in time and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>O</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi>h</m:mi> <m:mrow> <m:mi>k</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> O(h^{k+1}) accuracy in space in the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> L^{2} norm, and the numerical results are consistent with the theoretical conclusions. This indicates that the method can significantly reduce the degrees of freedom and save CPU time while maintaining high accuracy.