Success and failure of attempts to improve the accuracy of Raviart–Thomas mixed finite elements in curved domains

Abstract

Several important problems in Mechanics can be efficiently solved using Raviart–Thomas mixed finite element methods. Whenever the domain of interest has a curved boundary the methods of this family for N-simplexes are the natural choice. But in this case the question arises on the best way to prescribe normal flux conditions across the boundary, if any. It is generally acknowledged that the normal component of the flux variable should preferably not take up corresponding prescribed values at nodes shifted to the boundary of the approximating polytope in the underlying normal direction. This is because an accuracy downgrade is to be expected, as shown in Bertrand and Starke (2016). In that work an order-preserving technique was studied, based on a parametric version of these elements with curved simplexes. In this work an alternative with straight-edged triangles for two-dimensional problems is examined. The key feature of this approach is a Petrov–Galerkin formulation, in which the test-flux space is a little different from the shape-flux space. Based on previous author’s experience with this technique, as applied to Lagrange finite elements, it would lead to an overall accuracy improvement here as well. The experimentation reported hereafter provides examples and counterexamples confirming or not such an expectation, depending on the unknown field of the mixed problem at hand.