Open Access

ARTICLE

Stability and Error Analysis of Reduced-Order Methods Based on POD with Finite Element Solutions for Nonlocal Diffusion Problems

by Haolun Zhang¹, Mengna Yang¹, Jie Wei², Yufeng Nie^2,*

1 School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710129, China
2 Xi’an Key Laboratory of Scientific Computation and Applied Statistics, Northwestern Polytechnical University, Xi’an, 710129, China

* Corresponding Author: Yufeng Nie. Email:

(This article belongs to the Special Issue: Advances in Methods of Computational Modeling in Engineering Sciences, a Special Issue in Memory of Professor Satya Atluri)

Digital Engineering and Digital Twin 2024, 2, 49-77. https://doi.org/10.32604/dedt.2023.044180

Received 23 July 2023; Accepted 05 December 2023; Issue published 31 January 2024

Abstract

This paper mainly considers the formulation and theoretical analysis of the reduced-order numerical method constructed by proper orthogonal decomposition (POD) for nonlocal diffusion problems with a finite range of nonlocal interactions. We first set up the classical finite element discretization for nonlocal diffusion equations and briefly explain the difference between nonlocal and partial differential equations (PDEs). Nonlocal models have to handle double integrals when using finite element methods (FEMs), which causes the generation of algebraic systems to be more challenging and time-consuming, and discrete systems have less sparsity than those for PDEs. So we establish a reduced-order model (ROM) for nonlocal diffusion equations to alleviate the calculation load and expedite the solving process. The ROM is constructed using FE solutions in a small time interval as snapshots and has much fewer degrees of freedom than FEMs. We focus on discussing the existence, stability, and error estimates of the reduced-order solutions, which have not been considered in previous research for nonlocal models. Several numerical examples are presented to validate the theoretical conclusions and to show that the ROM is quite effective for solving nonlocal equations. Moreover, we systematically explore the effect of different parameters on the behavior of the POD algorithms. Both theoretical and experimental results offer valuable insights for developing more reliable and efficient algorithms.

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Keywords

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