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Stability and convergence of the mark and cell finite difference scheme for Darcy‐Stokes‐Brinkman equations on non‐uniform grids
Author(s) -
Sun Yue,
Rui Hongxing
Publication year - 2019
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22311
Subject(s) - mathematics , discretization , norm (philosophy) , mathematical analysis , convergence (economics) , uniform convergence , stability (learning theory) , perturbation (astronomy) , rate of convergence , physics , computer science , channel (broadcasting) , computer network , bandwidth (computing) , quantum mechanics , machine learning , political science , law , economics , economic growth
In this paper, we consider the mark and cell (MAC) method for Darcy‐Stokes‐Brinkman equations and analyze the stability and convergence of the method on nonuniform grids. Firstly, to obtain the stability for both velocity and pressure, we establish the discrete inf‐sup condition. Then we introduce an auxiliary function depending on the velocity and discretizing parameters to analyze the super‐convergence. Finally, we obtain the second‐order convergence in L2 norm for both velocity and pressure for the MAC scheme, when the perturbation parameter ϵ is not approaching 0. We also obtain the second‐order convergence for some terms of ∥·∥ ϵ norm of the velocity, and the other terms of ∥·∥ ϵ norm are second‐order convergence on uniform grid. Numerical experiments are carried out to verify the theoretical results.