Numerische Mathematik

, Volume 141, Issue 4, pp 1009–1042 | Cite as

Analysis of fully discrete FEM for miscible displacement in porous media with Bear–Scheidegger diffusion tensor

  • Wentao Cai
  • Buyang Li
  • Yanping Lin
  • Weiwei SunEmail author


Fully discrete Galerkin finite element methods are studied for the equations of miscible displacement in porous media with the commonly-used Bear–Scheidegger diffusion–dispersion tensor:
$$\begin{aligned} D(\mathbf{u}) = \gamma d_m I + |\mathbf{u}|\bigg ( \alpha _T I + (\alpha _L - \alpha _T) \frac{\mathbf{u} \otimes \mathbf{u}}{|\mathbf{u}|^2}\bigg ) \, . \end{aligned}$$
Previous works on optimal-order \(L^\infty (0,T;L^2)\)-norm error estimate required the regularity assumption \(\nabla _x\partial _tD(\mathbf{u}(x,t)) \in L^\infty (0,T;L^\infty (\Omega ))\), while the Bear–Scheidegger diffusion–dispersion tensor is only Lipschitz continuous even for a smooth velocity field \(\mathbf{u}\). In terms of the maximal \(L^p\)-regularity of fully discrete finite element solutions of parabolic equations, optimal error estimate in \(L^p(0,T;L^q)\)-norm and almost optimal error estimate in \(L^\infty (0,T;L^q)\)-norm are established under the assumption of \(D(\mathbf{u})\) being Lipschitz continuous with respect to \(\mathbf{u}\).



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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Wentao Cai
    • 1
  • Buyang Li
    • 2
  • Yanping Lin
    • 2
  • Weiwei Sun
    • 3
    • 4
    Email author
  1. 1.Department of Mathematics, School of SciencesHangzhou Dianzi UniversityHangzhouPeople’s Republic of China
  2. 2.Department of Applied MathematicsThe Hong Kong Polytechnic UniversityHong KongPeople’s Republic of China
  3. 3.Department of MathematicsCity University of Hong KongHong KongPeople’s Republic of China
  4. 4.School of Mathematical ScienceSouth China Normal UniversityGuangzhouPeople’s Republic of China

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