Least Squares for the Perturbed Stokes Equations and the Reissner--Mindlin Plate

Abstract

Abstract. In this paper, we develop two least-squares approaches for the solution of the Stokes equations perturbed by a Laplacian term. (Such perturbed Stokes equations arise from finite element approximations of the Reissner-Mindlin plate.) Both are two-stage algorithms that solve first for the curls of the rotation of the fibers and the solenoidal part of the shear strain, then for the rotation itself (if desired). One approach uses L 2 norms and the other approach uses H −1 norms to define the least-squares functionals. It is shown that the H −1 norm approach, under general assumptions, and the L 2 norm approach, under certain H 2 regularity assumptions, admit optimal performance for standard finite element discretization and either standard multigrid solution methods or preconditioners. These methods do not degrade when the perturbed parameter (the plate thickness) approaches zero. We also develop a three-stage least-squares method for the Reissner-Mindlin plate, which first solves for the curls of the rotation and the shear strain, next for the rotation itself, and then for the transverse displacement.

A DPG method for Reissner-Mindlin plates

“…In the case of U (t), p ∈ L 2 (Ω) R is unique only up to an additive constant, cf. (16). In practice, this can be fixed by adding a rank-one term to the linear system to require (p , 1) = 0.…”

“…We also note that DPG schemes, being of minimum residual type, are related with least squares approaches. Such methods have been studied for a perturbed Stokes problem which behaves like the Reissner-Mindlin model, see [17,16]. This list is far from being complete.…”

Reconstructed Discontinuous Approximation to Stokes Equation in a Sequential Least Squares Formulation

“…Bochev and Gunzburger developed a least squares approach based on rewriting the velocity-vorticitypressure formulation as a first-order elliptic system [8]. Cai and his coworkers developed the least squares finite element method based on the L 2 norm residual and C 0 spaces for the Stokes problem, we refer to [5,[14][15][16] for more details. Liu et al developed a hybrid least squares finite element method based on continuous finite element spaces.…”

Reconstructed Discontinuous Approximation to Stokes Equation in A Sequential Least Squares Formulation

“…Bochev and Gunzburger developed a least squares approach based on rewriting the velocity-vorticity-pressure formulation as a first-order elliptic system [8]. Cai and his coworkers developed the least squares finite element method based on the L 2 norm residual and C 0 spaces for the Stokes problem, we refer to [15,5,16,14] for more details. Liu et al developed a hybrid least squares finite element method based on continuous finite element spaces.…”