Three iterative methods of Darcy–Brinkman equations based on the finite element method

Peixin Sun; Lunji Song; Youlei Liang; Peixin Sun; Lunji Song; Youlei Liang

doi:10.3934/math.20261226

AIMS Mathematics

2026, Volume 11, Issue 9: 30979-31004. doi: 10.3934/math.20261226

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Three iterative methods of Darcy–Brinkman equations based on the finite element method

School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China

Received: 10 April 2026 Revised: 24 July 2026 Accepted: 20 August 2026 Published: 21 September 2026
MSC : 35Q35, 65N30, 76S05

In this paper, three iterative methods—Stokes, Newton, and Oseen iterations—are applied to solve the nonlinear coupled system resulting from the finite element discretization of Darcy–Brinkman equations. The three iterative methods reduce the nonlinear problem to a sequence of linear subproblems, enhancing computational efficiency while maintaining accuracy. Error estimates and convergence are derived for the three iterative methods within the finite element framework. Numerical experiments are conducted to evaluate and compare the convergence rates of the three iterative methods, validating the theory.
- steady Darcy–Brinkman equations,
- Stokes iterative method,
- Newton iterative method,
- Oseen iterative method,
- finite element method
Citation: Peixin Sun, Lunji Song, Youlei Liang. Three iterative methods of Darcy–Brinkman equations based on the finite element method[J]. AIMS Mathematics, 2026, 11(9): 30979-31004. doi: 10.3934/math.20261226

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Abstract

In this paper, three iterative methods—Stokes, Newton, and Oseen iterations—are applied to solve the nonlinear coupled system resulting from the finite element discretization of Darcy–Brinkman equations. The three iterative methods reduce the nonlinear problem to a sequence of linear subproblems, enhancing computational efficiency while maintaining accuracy. Error estimates and convergence are derived for the three iterative methods within the finite element framework. Numerical experiments are conducted to evaluate and compare the convergence rates of the three iterative methods, validating the theory.

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