A nonlinear weighted least-squares finite element method for Stokes equations

Abstract

a b s t r a c tThe paper concerns a nonlinear weighted least-squares finite element method for the solutions of the incompressible Stokes equations based on the application of the leastsquares minimization principle to an equivalent first order velocity-pressure-stress system. Model problem considered is the flow in a planar channel. The least-squares functional involves the L 2 -norms of the residuals of each equation multiplied by a nonlinear weighting function and mesh dependent weights. Using linear approximations for all variables, by properly adjusting the importance of the mass conservation equation and a carefully chosen nonlinear weighting function, the least-squares solutions exhibit optimal L 2 -norm error convergence in all unknowns. Numerical solutions of the flow pass through a 4 to 1 contraction channel will also be considered.

A nonlinear weighted least-squares finite element method for the Carreau–Yasuda non-Newtonian model

“…In (22), the weight h 2 and the function w s are chosen based on similar considerations as those used in [13,14]. The weight w s in (22) indicates that the nonlinear weight w s is evaluated at u 0 .…”

“…Least-squares finite element methods have been reported to offer several theoretical and computational advantages over the Galerkin method for various boundary value problems [2]. Discretization generates an algebraic system that is always symmetric and positive definite, and a single approximating space for all variables can be used for programming least-squares finite element methods [14]. The least-squares functional of the velocity-pressure-stress formulation has the advantage that stress tensor components are computed directly [13].…”

“…In [14], Lee and Chen propose a nonlinear weighted least-squares (NL-WDLS) method that allows for the use of simple combinations of interpolations, including equal-order linear elements for Stokes equations. They indicate the choice of weights used to balance the residual contributions, and their results show some improvement over the case with no weightings.…”

Decoupled algorithm for solving Phan-Thien–Tanner viscoelastic fluid by finite element method

“…In recent years, the least-squares finite element method has been developed for many applications in fluid mechanics [7][8][9][10][11][12][13]. In [7], Bochev and Gunzburger gave a weighted least-squares finite element method for an equivalent first order Stokes system.…”

“…In [7], Bochev and Gunzburger gave a weighted least-squares finite element method for an equivalent first order Stokes system. The least-squares finite element method has also been applied to the Oldroyd-B, Carreau, Giesekus and upperconvected Maxwell models for numerical simulations, see e.g., [4,[11][12][13]. The least-squares finite element method is based on the minimization of a quadratic functional including the residuals of each equation multiplied by a proper weight.…”