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

SummaryThis 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→Ωh) 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 overDh, a union of elements, with approximation powerhk in theL2 norm and with dist (Ω, Ωh)≦Chk, one obtains optimal rates of convergence in theH1 andL2 norms whether Ωh is fitted (Ωh≡Dh) or unfitted (Ωh⊂Dh) provided the numerical integration scheme has sufficient accuracy.