A discontinuous Petrov–Galerkin methodology for adaptive solutions to the incompressible Navier–Stokes equations

Abstract

The discontinuous Petrov–Galerkin methodology with optimal test functions (DPG) of Demkowicz and Gopalakrishnan [18,20] guarantees the optimality of the solution in an energy norm, and provides several features facilitating adaptive schemes. Whereas Bubnov–Galerkin methods use identical trial and test spaces, Petrov–Galerkin methods allow these function spaces to differ. In DPG, test functions are computed on the fly and are chosen to realize the supremum in the inf–sup condition; the method is equivalent to a minimum residual method. For well-posed problems with sufficiently regular solutions, DPG can be shown to converge at optimal rates—the inf–sup constants governing the convergence are mesh-independent, and of the same order as those governing the continuous problem [48]. DPG also provides an accurate mechanism for measuring the error, and this can be used to drive adaptive mesh refinements.We employ DPG to solve the steady incompressible Navier–Stokes equations in two dimensions, building on previous work on the Stokes equations, and focusing particularly on the usefulness of the approach for automatic adaptivity starting from a coarse mesh. We apply our approach to a manufactured solution due to Kovasznay as well as the lid-driven cavity flow, backward-facing step, and flow past a cylinder problems.