A practical finite element approximation of a semi-definite Neumann problem on a curved domain

Summary

This paper considers the finite element approximation of the semi-definite Neumann problem: −∇·(σ∇u)=f in a curved domain Ω⊂ℝⁿ (n=2 or 3),$\sigma \frac{{\partial u}}{{\partial v}} = g$ on πΩ and$\int\limits_\Omega {u dx} = q$, a given constant, for dataf andg satisfying the compatibility condition$\int\limits_\Omega {f dx} + \int\limits_{\partial \Omega } {g ds} = 0$. Due to perturbation of domain errors (Ω→Ω^h) the standard Galerkin approximation to the above problem may not have a solution. A remedy is to perturb the right hand side so that a discrete form of the compatibility condition holds. Using this approach we show that for a finite element space defined overD ^h, a union of elements, with approximation powerh ^k in theL ² norm and with dist (Ω, Ω^h)≦Ch ^k, one obtains optimal rates of convergence in theH ¹ andL ² norms whether Ω^h is fitted (Ω^h≡D ^h) or unfitted (Ω^h⊂D ^h) provided the numerical integration scheme has sufficient accuracy.