A variant of the Raviart-Thomas method for smooth domains with straight-edged triangles

TL;DR

This paper introduces a new straight-edged triangle variant of the Raviart-Thomas mixed finite element method for 2D elliptic problems, ensuring stability and optimal convergence, especially for Neumann boundary conditions on smooth domains.

Contribution

It proposes a Petrov-Galerkin formulation with straight-edged triangles, improving flux approximation near boundaries compared to previous curved-simplex methods.

Findings

The method is uniformly stable.

It achieves optimal convergence rates.

Effective for problems with Neumann boundary conditions.

Abstract

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 previous papers by the first author et al. In that work an order-preserving technique was studied, based on a parametric version of these elements with curved simplexes. In this article 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 a uniformly stable and optimally convergent method in the natural norm, taking the Poisson equation as a model problem.