Off-site enforcement of natural conditions on smooth boundaries for finite elements upon fitted straight-edged triangular meshes

A few decades ago some possible remedies to an inaccurate enforcement of Neumann or Robin conditions prescribed on the boundary of a smooth domain, owing to the approximation of a curved domain by the union of straight-edged triangles or tetrahedra in a fitted mesh, were addressed in the literature.

By that time authors such as Barrett and Elliott (1988) advocated the use of elements with a single curved edge or face fitting the true boundary not only at two or three vertexes, but also at additional points on those curves or curved surfaces, so as to define a polynomial surface of a certain type compatible with the theoretical approximation order of the method in use.

In this work we adopt a different approach, whose main feature is the use of a fitted mesh consisting of straight-edged elements only.

The recovery of lost accuracy due to the domain's approximation by a polytope is achieved by means of the addition of terms to the bilinear form, which account for natural boundary conditions of the same type to be prescribed on the approximating boundary, though much closer to the true ones.

This technique is applied here to the case of triangular Lagrange finite elements, for which we give a rigorous reliability study in the solution of reaction-diffusion equations.

Numerical experimentation is supplied in support of the theoretical results.