A variant of the Raviart-Thomas method to handle smooth domains using straight-edged triangles. Part II -- Approximation results

Abstract: In arXiv:2307.03503 [math.NA] we commenced to study a variant of the Raviart-Thomas mixed finite element method for triangles, to solve second order elliptic equations in a curved domain with Neumann or mixed boundary conditions. It is well known that in such a case the normal component of the flux variable should 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 previous work by the first author et al. An order-preserving technique was studied therein, based on a parametric version of these elements with curved simplexes. Our variant is an alternative to the approach advocated in those articles, allowing to achieve the same effect with straight-edged triangles. 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. In this paper we first recall the description of this method, together with underlying uniform stability results given in arXiv:2307.03503 [math.NA]. Then we show that it gives rise to optimal-order interpolation in the space H(div). Accordingly a priori error estimates are obtained for the Poisson equation taken as a model.