Finite element and isogeometric stabilized methods for the advection-diffusion-reaction equation

We develop two new stabilized methods for the steady advection-diffusion-reaction equation, referred to as the Streamline GSC Method and the Directional GSC Method. Both are globally conservative and perform well in numerical studies utilizing linear, quadratic, cubic, and quartic Lagrange finite elements and maximally smooth B-spline elements. For the streamline GSC method we can prove coercivity, convergence, and optimal-order error estimates in a strong norm that are robust in the advective and reactive limits. The directional GSC method is designed to accurately resolve boundary layers for flows that impinge upon the boundary at an angle, a long-standing problem. The directional GSC method performs better than the streamline GSC method in the numerical studies, but it is not coercive. We conjecture it is inf-sup stable but we are unable to prove it at this time. However, calculations of the inf-sup constant support the conjecture. In the numerical studies, B-spline finite elements consistently perform better than Lagrange finite elements of the same order and number of unknowns.