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Discrete mass conservation for porous media saturated flow
Author(s) -
Jenkins Eleanor W.,
Paribello Chris,
Wilson Nicholas E.
Publication year - 2014
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.21831
Subject(s) - porous medium , conservation of mass , pointwise , mathematics , flow (mathematics) , convergence (economics) , conservation law , finite element method , partial differential equation , divergence (linguistics) , mathematical analysis , calculus (dental) , porosity , mechanics , geometry , thermodynamics , physics , geology , geotechnical engineering , economic growth , economics , medicine , dentistry , linguistics , philosophy
Global and local mass conservation for velocity fields associated with saturated porous media flow have long been recognized as integral components of any numerical scheme attempting to simulate these flows. In this work, we study finite element discretizations for saturated porous media flow that use Taylor–Hood (TH) and Scott–Vogelius (SV) finite elements. The governing equations are modified to include a stabilization term when using the TH elements, and we provide a theoretical result that shows convergence (with respect to the stabilization parameter) to pointwise mass‐conservative solutions. We also provide results using the SV approximation pair. These elements are pointwise divergence free, leading to optimal convergence rates and numerical solutions. We give numerical results to verify our theory and a comparison with standard mixed methods for saturated flow problems. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 625–640, 2014