A variant of the Raviart–Thomas method to handle smooth domains using straight-edged triangles

Several physical problems modeled by second-order elliptic equations can be efficiently solved using mixed finite elements of the Raviart–Thomas family RTk for N-simplexes, introduced in the seventies. In case Neumann conditions are prescribed on a curvilinear boundary, the normal component of the flux variable should preferably not take up values at nodes shifted to the boundary of the approximating polytope in the corresponding normal direction. This is because the method’s accuracy downgrades, which was shown in Bertrand and Starke (SIAM J. Numer. Anal. 54 (2016) 3648–3667). In that work an order-preserving technique was studied, based on a parametric version of these elements with curved simplexes. In this paper an alternative with straight-edged triangles for two-dimensional problems is proposed. The key point of this method is a Petrov–Galerkin formulation of the mixed problem, in which the test-flux space is a little different from the shape-flux space. After describing the underlying variant of RTk, we show that it gives rise to uniformly stable methods taking the Poisson equation as a model problem. Then a priori error estimates of optimal order in the [H(div) × L2]-norm are demonstrated.