Resumen

We introduce a convergence analysis of the finite element method applied to the Reynolds-Orr eigenvalue problem in wall-bounded shear-driven incompressible flows with arbitrary cross-section. The Reynolds-Orr eigenproblem can be written as a mixed formulation similar to Stokes flow, but including an extra term involving the strain rate tensor of the underlying laminar flow. The analysis of the resulting discrete eigenproblem must be adapted to the standard spectral approximation framework, since one of the bilinear forms which is coercive in the Stokes equations is no longer coercive. We demonstrate that the proposed approach delivers accurate estimates of errors associated with both eigenvalues and eigenfunctions. We carry out various numerical tests to showcase how well the method performs and to confirm the accuracy of our theoretical results.