×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Babu?ka, I., Aziz, A.K.: Survey lectures on the mathematical foundations of the finite element method. In: Aziz, A.K. (ed.), The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, pp. 3-363. New York: Academic Press 1972 · Zbl 0268.65052
[2] Barrett, J.W., Elliott, C.M.: A finite element method for solving elliptic equations with Neumann data on a curved boundary using unfitted meshes. IMAJ Numer. Anal.4, 309-325 (1984a) · Zbl 0574.65121
[3] Barrett, J.W., Elliott, C.M.: Total flux estimates for a finite element approximation of elliptic equations. IMAJ Numer. Anal. (to appear) · Zbl 0619.65098
[4] Barrett, J.W., Elliott, C.M.: Finite element approximation of elliptic equations with a Neumann or Robin condition on a curved boundary. IMAJ Numer. Anal. (submitted) · Zbl 0658.65106
[5] Ciarlet, P.G., Raviart, P.A.: The combined effect of curved boundaries and numerical integration in isoparametric finite element methods. In: Aziz, A.K. (ed.), The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, pp. 409-474. New York: Academic Press 1972 · Zbl 0262.65070
[6] Ferris, D.H., Martin, D.W.: Numerical solution of discrete Poisson-Neumann problems with compatible or incompatible data, with reference to flow in a circular cavity. IMAJ Numer. Anal.5, 79-100 (1985) · Zbl 0631.76028
[7] Forsythe, G.E., Wasow, W.R.: Finite Difference Methods for Partial Differential Equations. New York: Wiley 1960 · Zbl 0099.11103
[8] Kufner, A., John, O., Fucik, S.: Function Spaces. Leyden: Nordhoff 1977
[9] Molchanov, I.N., Galba, E.F.: On finite element methods for the Neumann problem. Numer. Math.46, 587-598 (1985) · Zbl 0571.65099
[10] Ne?as, J.: Les M?thodes Directes en Th?orie des Equations Elliptiques. Paris: Masson 1967
[11] Nedoma, J.: The finite element solution of elliptic and parabolic equations using simplicial isoparametric elements. RAIRO Anal. Numer.13, 257-289 (1979) · Zbl 0413.65080
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.