DOI: 10.1515/cmam-2025-0194 ISSN: 1609-4840

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

Zihan Zheng, Fuzheng Gao, Jintao Cui
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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

O ⁢ ( h k + 1 ) O(h^{k+1})
accuracy in space in the
L 2 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.

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