We derive stability properties and error estimates for the finite element method when used to approximate heat flow in a fluid enclosed by a solid medium. The coupled Navier Stokes system involves the Boussinesq equations in the fluid-filled cavity linked through an interface with heat conduction in the solid enclosing the fluid. As we assume no extra regularity then can be shown to hold under mild restriction on the data (at least over a small time interval in R3), we focus primarily on low order finite element spaces.
Summary. An analysis of the Babuw stability of bilinear/constant finite element pairs for viscous flow calculations is given. An unstable mode not of the checkerboard type is given for which the stability constant turns out to be O(h). Thus, the indicated spaces are not stable in general for numerical calculation.
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