A least-squares finite element method for incompressible flow in stress-velocity-pressure version

Finite Element Methods of Least-Squares Type

“…Elimination of the velocity field in (3.52) can be considered as an artifact since one can simply consider (3.52) together with (3.54). Such a first-order system is studied in [53], where the new variables are called "stresses" and the corresponding first-order system is called the "stress-velocity-pressure" Stokes system despite the fact that the new variables are not the components of the stress tensor. This is not to be confused with the formulation of §3.2.2 for which the true stresses are used.…”

“…The analysis and implementation of such methods have drawn most of the attention of researchers interested in modern least-squares finite element methods and there exists an abundant mathematical and engineering literature devoted to this subject; see, e.g., [3], [9], [11]- [19], [25], [34], [36], [47], [54], [57], [50], [51], [53], [59], [68], [87], [90]- [94], [96], [99], [105], [112], and [117] among others. As a result, least-squares finite element methods in these settings are among the best understood, studied, and tested from both the theoretical and computational viewpoints.…”

Least-squares methods for the velocity-pressure-stress formulation of the Stokes equations

“…Indeed, if the least-squares functional contains only L 2 -norms of the residuals of the equations and if the highest order of differentiation in each equation does not exceed one then it is possible to obtain conforming discretizations of the variational problem (the Euler-Lagrange equation) using simple continuous finite element spaces. So far this approach has been applied succesfully to the Stokes equations cast into first order systems involving the velocity, vorticity and the pressure, [5], [7], [10], [13] and the accleration and the pressure [9] as the dependent variables.…”

“…[4], [5], [6], [7], [9], [10], [12], [13], [19], [20], [21], [22], [24]. In contrast with the mixed methods, the variational problem in the least-squares approach is derived as the necessary condition (Euler-Lagrange equation) for the minimizers of a suitably defined quadratic functional involving residuals of the differential equations.…”

Analysis of least-squares mixed finite element methods for nonlinear nonstationary convection-diffusion problems

“…Recently, there has been an increasing interest in applications of least-squares finite element algorithms to various problems, steady or evolutionary. Many papers have been written on applications and theories of least-squares finite element methods for various elliptic boundary value problems, e.g., see [2,3,7,8,10,11,12,13,14,18,19,20,21,25,26]. In recent years, least-squares finite element methods have been extended to many nonstationary problems, e.g., see [9,14,15,16,24,28].…”