<i>p</i>‐version least squares finite element formulation for two‐dimensional, incompressible fluid flow

Abstract

Abstract A p ‐version least squares finite element formulation for non‐linear problems is applied to the problem of steady, two‐dimensional, incompressible fluid flow. The Navier‐Stokes equations are cast as a set of first‐order equations involving viscous stresses as auxiliary variables. Both the primary and auxiliary variables are interpolated using equal‐order C 0 continuity, p ‐version hierarchical approximation functions. The least squares functional (or error functional) is constructed using the system of coupled first‐order non‐linear partial differential equations without linearization, approximations or assumptions. The minimization of this least squares error functional results in finding a solution vector {δ} for which the partial derivative of the error functional (integrated sum of squares of the errors resulting from individual equations for the entire discretization) with respect to the nodal degrees of freedom {δ} becomes zero. This is accomplished by using Newton's method with a line search. Numerical examples are presented to demonstrate the convergence characteristics and accuracy of the method.