A Galerkin Finite Element Method for a Nonlocal Parabolic System with Nonlinear Boundary Conditions Arising from the Thermal Explosion Theory

Qipeng Guo, Yu Zhang and Baoqiang Yan ()
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Qipeng Guo: School of Mathematics and Statistics, Shandong Normal University, Jinan 250358, China
Yu Zhang: School of Mathematics and Statistics, Shandong Normal University, Jinan 250358, China
Baoqiang Yan: School of Mathematics and Statistics, Shandong Normal University, Jinan 250358, China

Mathematics, 2025, vol. 13, issue 5, 1-22

Abstract: In this paper, we discuss a class of nonlocal parabolic systems with nonlinear boundary conditions arising from the thermal explosion theory. First, we prove the local existence and uniqueness of the classical solution using the Leray–Schauder fixed-point theorem. Then, we analyze three Galerkin approximations of the system and derive the optimal-order error estimates: O ( h r + 1 ) in L 2 norm for continuous-time Galerkin approximation, O ( h r + 1 + ( Δ t ) 2 ) in the L 2 norm for Crank–Nicolson Galerkin approximation, and O ( h r + 1 + ( Δ t ) 2 ) in both L 2 and H 1 norms for extrapolated Crank–Nicolson Galerkin approximation.

Keywords: Galerkin finite element method; nonlocal parabolic system; fixed-point theorem; nonlinear boundary conditions; uniqueness; error estimate (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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