Abstract
We use Fourier analysis to investigate the instability of an equal-order mixed finite element approximation method for elliptic incompressible flow equations. The lack of stability can be attributed to the fact that the associated discrete Ladyzhenskaya--Babuska--Brezzi (LBB) constant tends to zero as the mesh size is reduced. We develop a stabilization approach that is appropriate to the periodic setting and deduce optimal choices of the associated stabilization parameter.
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