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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

##### MSC:
 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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##### References:
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