New Numerical Approach for the Steady-State Navier–Stokes Equations with Corner Singularity

Abstract

The steady-state nonlinear problem governing the Newtonian flow of an incompressible viscous fluid in convection form in L-shaped domain is considered. The solution as [Formula: see text]-generalized one in weighted sets is determined. Unlike the classical definition of the generalized solution in a weak formulation, which has a symmetric structure, our setting has an asymmetric one so that a special inf-sup condition in weighted sets is proved. A new weighted finite element scheme for an approximate [Formula: see text]-generalized solution is constructed. A converging iterative method to solve the sequence of linear problems with block preconditioning of their matrices is introduced. The results of numerical simulations of several tests have shown an advantage over the standard methods so that an approximate [Formula: see text]-generalized solution tends to the exact one with a rate almost twice in order exceed, relative to the grid step, than classical approaches. The result is achieved without using mesh refined in the neighborhood of a reentrant corner.