Finite-Element Approximation of Elliptic Equations with a Neumann or Robin Condition on a Curved Boundary

This paper considers a finite-element approximation of a second-order self adjoint elliptic equation in a region Ω⊂Rⁿ (with n=2 or 3) having a curved boundary ∂Ω on which a Neumann or Robin condition is prescribed. If the finite-element space defined over $D¯h$, a union of elements, has approximation power h^k in the L² norm, and if the region of integration is approximated by Ω^h with dist (Ω, Ω^h)≤Ch^k, then it is shown that one retains optimal rates of convergence for the error in the H¹ and L² norms, whether Q^h is fitted $Ω¯h=D¯h$ or unfitted $Ω¯h⊂D¯h$, provided that the numerical integration scheme has sufficient accuracy.