Abstract
In this paper, we discuss numerical methods to solve a coupled Stokes–Darcy system. The deformation tensor form of Stokes equations is used to describe the fluid flow motion. The mixed form of elliptic equation is applied to describe the porous media flow motion. We propose finite element methods for the coupled problem. For Stokes equations, one component of the velocity is approximated by Crouzeix–Raviart element or Rannacher–Turek element, and the other component is approximated by conforming P 1 or Q 1 element; pressure is approximated by piecewise constants. For the mixed form of elliptic equation, the lowest order triangular/quadrilateral Raviart-Thomas element is used. The discrete mesh is nonmatching on the interface. By Boland-Nicolaides trick, the inf-sup condition of the discrete problem is proved. Moreover, we construct a new interpolation operator to derive the a priori error estimate of the proposed finite element method. Numerical examples are also given to confirm the theoretical results.
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This work was supported by the National Natural Science Foundation of China under Grant 11071124, 11226309, 11371199, 11301267 and the Natural Science Foundation for Colleges and Universities in Jiangsu Province under Grant 12KJB110007 and the Jiangsu Planned Projects for Postdoctoral Research Funds under Grant 1202086C.
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Huang, P., Chen, J. Some low order nonconforming mixed finite elements combined with Raviart–Thomas elements for a coupled Stokes–Darcy model. Japan J. Indust. Appl. Math. 30, 565–584 (2013). https://doi.org/10.1007/s13160-013-0119-z
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DOI: https://doi.org/10.1007/s13160-013-0119-z
Keywords
- Mortar method
- Nonconforming element
- Mixed element
- Crouzeix–Raviart element
- Rotated Q 1 element
- Raviart–Thomas element
- Korn’s inequality
- Inf-sup condition
- Stokes equations
- Darcy’s equations
- Error estimate