Compatible Spectral Approximations for the Velocity-Pressure-Stress Formulation of the Stokes Problem

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Abstract
The numerical approximation of the mixed velocity-pressure-stress formulation of the Stokes problem using spectral methods is considered. In addition to the compatibility condition between the discrete velocity and pressure spaces, a second condition between the discrete velocity and stress spaces must also be satisfied in order to have a well-posed problem. The theory is developed by considering a doubly constrained minimization problem in which the viscous stress tensor is minimized subject to the constraint that the viscous forces are irrotational. The discrete problem is analyzed and error estimates are derived. A comparison between the mixed approach and the standard velocity-pressure formulation is made in terms of the condition number of the resulting systems.

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