A New Divergence-Free Interpolation Operator with Applications to the Darcy-Stokes-Brinkman Equations

Authors:
Xuejun Xu;Shangyou Zhang
Affiliations:
xxj@lsec.cc.ac.cn;szhang@udel.edu
Venue:
SIAM Journal on Scientific Computing
Year:
2010

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A Mixed and Nonconforming FEM with Nonmatching Meshes for a Coupled Stokes-Darcy Model

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Abstract

A new local interpolation operator, preserving the divergence, is constructed explicitly for the Hsieh-Clough-Tocher divergence-free element. A divergence-free finite element method is applied to the Darcy-Stokes-Brinkman flow in a mixed region of both free and porous media. The method is of optimal order as well for the Darcy flow as for the Stokes flow (which is not the case for most other finite elements). Compared to the existing nonconforming elements, the divergence-free element method provides a continuous solution for the velocity which is also an orthogonal projection within a Hilbert subspace of the true velocity. Numerical tests supporting the theory are presented.