A high order term-by-term stabilization solver for incompressible flow problems

In this paper, we introduce a low-cost, high-order stabilized method for the numerical solution of incompressible flow problems. This is a particular type of projection-stabilized method where each targeted operator, such as the pressure gradient or the convection, is stabilized by least-squares terms added to the Galerkin formulation. The main methodological originality is that we replace the projection-stabilized structure by an interpolation-stabilized structure, with reduced computational cost for some choices of the interpolation operator. This stabilization has one level, in the sense that it is defined on a single mesh. We prove the stability of our formulation by means of a specific inf-sup condition, which is the main technical innovation of our paper. We perform a convergence and error estimates analysis, proving the optimal order of accuracy of our method. We include some numerical tests that confirm our theoretical expectations.