Least squares for the perturbed Stokes equations and the Reissner--Mindlin plate (Q2706377)

The purpose is to develop two least-squares approaches directly approximating the rotation flux for the Stokes equation perturbed by Laplacian term which arises from finite element approximation of Reissner-Mindlin plate. Both methods are two-stage algorithms that solve first the curls of rotation of fibers and solenoidal part of the shear strain, and then solve the rotation itself (if desired). These methods do not degrade when the perturbed parameter (the plate thickness) approaches zero. One approach uses \(L^2\) norms to define the least-square functional, and it yields uniform and optimal \(H^1\) approximations of all variables under certain \(H^2\) regularity assumptions. The other approach is based on the least-squares functional in \(H^{-1}\) norm, and yields uniform and optimal \(L^2\) approximations for the rotation flux and \(H^1\) approximations for the pressure.NEWLINENEWLINENEWLINEThe author also develops a direct approach for Reissner-Mindlin plate. Such a method solves for the rotation flux and transverse shear strain variables first. The rotation components and the transverse displacements component can then be obtained by solving Poisson equations (if desired).