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A practical finite element approximation of a semi-definite Neumann problem on a curved domain. (English) Zbl 0617.65110
In this very interesting paper the authors consider a finite element approximation of the semi-definite Neumann problem: \(-\nabla \cdot (\sigma \nabla u)=f\) in a curved domain \(\Omega \subset {\mathbb{R}}^ n\) \((n=2\) or 3), \(\sigma (\partial u/\partial v)=g\) on \(\partial \Omega\) and \(\int_{\Omega}udx=q\), a given constant, for data f and g satisfying the compatibility condition \(\int_{\Omega}fdx+\int_{\partial \Omega}gds=0\). Due to perturbation of domain errors \((\Omega \to \Omega^ h)\) 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. The authors show that for a finite element space defined over \(D^ h\), a union of elements, with approximation power \(h^ k\) in the \(L^ 2\) norm and with \(dist(\Omega,\Omega^ h)\leq Ch^ k\), one obtains optimal rates of convergence in the \(H^ 1\) and \(L^ 2\) norms whether \(\Omega^ h\) is fitted \((\Omega^ h\equiv D^ h)\) or unfitted \((\Omega^ h\subset D^ h)\) provided the numerical integration scheme has sufficient accuracy.
Reviewer: P.Neittaanmäki

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
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