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Mixing traction/pseudo-traction boundary conditions in incompressible flow

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Old   January 10, 2014, 11:21
Default Mixing traction/pseudo-traction boundary conditions in incompressible flow
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I was wondering if there is a way to incorporate the traction boundary conditions in discretizations using the pseudo-traction form of the Stokes problem or vice versa. Is there some literature on this or does one simply have no chance? I am mainly interested in finite-elements, but pointers to work dealing with other discretizations or publications dealing with both types of boundary conditions within one simulation would also be helpful.

Stress form of Stokes with pseudo-traction b.c.:

-\nabla\cdot(\nu(\nabla \mathbf{u} + (\nabla \mathbf{u})^T) - p I) = \mathbf{f},  \quad \text{and} \quad  \nabla\cdot \mathbf{u} = 0 \quad \text{in} \quad \Omega

\mathbf{u} = 0 \quad \text{on} \quad \partial\Omega\backslash\Gamma,  \quad \text{and} \quad (\nu\nabla \mathbf{u} - p I)\cdot\mathbf{n} = \mathbf{t}\quad \text{on} \quad \Gamma

Pseudo-stress form of Stokes with traction b.c.:

-\nu\Delta\mathbf{u} + \nabla p = \mathbf{f},  \quad \text{and} \quad  \nabla\cdot \mathbf{u} = 0 \quad \text{in} \quad \Omega

\mathbf{u} = 0 \quad \text{on} \quad \partial\Omega\backslash\Gamma,  \quad \text{and} \quad (\nu(\nabla \mathbf{u} + (\nabla \mathbf{u})^T) - p I)\cdot\mathbf{n} = \mathbf{t}\quad \text{on} \quad \Gamma

I am aware that these boundary conditions are not natural in the respective (weak) formulations, but out of curiosity I would like to know if there are approaches to incorporate these boundary conditions anyway in a stable and convergent manner.
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