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

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Abstract

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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Authors and Affiliations

  1. Department of Mathematics, School of Sciences, Hangzhou Dianzi University, Hangzhou, 310018, People’s Republic of China

    Wentao Cai

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, People’s Republic of China

    Buyang Li & Yanping Lin

  3. Department of Mathematics, City University of Hong Kong, Hong Kong, People’s Republic of China

    Weiwei Sun

  4. School of Mathematical Science, South China Normal University, Guangzhou, People’s Republic of China

    Weiwei Sun

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  1. Wentao Cai
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Correspondence to Weiwei Sun.

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Wentao Cai: The research of this author was supported in part by a Hong Kong RGC Grant (15301818).

Buyang Li: The research of this author was supported in part by an internal grant of The Hong Kong Polytechnic University (project code 1-ZE6L) and a Hong Kong RGC Grant (15301818).

Yanping Lin: The research of this author was supported in part by a Hong Kong RGC Grant (15302418).

Weiwei Sun: The research of this author was supported in part by the Zhujiang Scholar program, a grant from South China Normal University and a Hong Kong RGC Grant (CityU 11300517).

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Cai, W., Li, B., Lin, Y. et al. Analysis of fully discrete FEM for miscible displacement in porous media with Bear–Scheidegger diffusion tensor. Numer. Math. 141, 1009–1042 (2019). https://doi.org/10.1007/s00211-019-01030-0

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