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Optimal error analysis of Crank–Nicolson lowest‐order Galerkin‐mixed finite element method for incompressible miscible flow in porous media
Author(s) -
Gao Huadong,
Sun Weiwei
Publication year - 2020
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.22503
Subject(s) - mathematics , finite element method , discontinuous galerkin method , galerkin method , polygon mesh , compressibility , incompressible flow , mathematical analysis , crank–nicolson method , norm (philosophy) , numerical analysis , flow (mathematics) , geometry , mechanics , physics , political science , law , thermodynamics
Numerical methods for incompressible miscible flow in porous media have been studied extensively in the last several decades. In practical applications, the lowest‐order Galerkin‐mixed method is the most popular one, where the linear Lagrange element is used for the concentration and the lowest order Raviart–Thomas mixed element pair is used for the Darcy velocity and pressure. The existing error estimate of the method in L 2 ‐norm is in the order Oh p + h c 2in spatial direction, which however is not optimal and valid only under certain extra restrictions on both time step and spatial meshes, excluding the most commonly used mesh h = h p = h c . This paper focuses on new and optimal error estimates of a linearized Crank–Nicolson lowest‐order Galerkin‐mixed finite element method (FEM), where the second‐order accuracy for the concentration in both time and spatial directions is established unconditionally. The key to our optimal error analysis is an elliptic quasi‐projection. Moreover, we propose a simple one‐step recovery technique to obtain a new numerical Darcy velocity and pressure of second‐order accuracy. Numerical results for both two and three‐dimensional models are provided to confirm our theoretical analysis.