A computational method for approximating a Darcy–Stokes system governing a vuggy porous medium |
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| Authors: | Todd Arbogast Dana S Brunson |
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| Institution: | (1) Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin, TX 78712, USA;(2) Institute for Computational Engineering and Sciences, The University of Texas at Austin, 1 University Station C0200, Austin, TX 78712, USA |
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| Abstract: | We develop and analyze a mixed finite element method for the solution of an elliptic system modeling a porous medium with
large cavities, called vugs. It consists of a second-order elliptic (i.e., Darcy) equation on part of the domain coupled to
a Stokes equation on the rest of the domain, and a slip boundary condition (due to Beavers–Joseph–Saffman) on the interface
between them. The tangential velocity is not continuous on the interface. We consider a 2-D vuggy porous medium with many
small cavities throughout its extent, so the interface is not isolated. We use a certain conforming Stokes element on rectangles,
slightly modified near the interface to account for the tangential discontinuity. This gives a mixed finite element method
for the entire Darcy–Stokes system with a regular sparsity pattern that is easy to implement, independent of the vug geometry,
as long as it aligns with the grid. We prove optimal global first-order L
2 convergence of the velocity and pressure, as well as the velocity gradient in the Stokes domain. Numerical results verify
these rates of convergence and even suggest somewhat better convergence in certain situations. Finally, we present a lower
dimensional space that uses Raviart–Thomas elements in the Darcy domain and uses our new modified elements near the interface
in transition to the Stokes elements. |
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| Keywords: | Beavers– Joseph boundary condition Darcy– Stokes system error estimates mixed finite elements vuggy porous media |
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