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Fictitious domain finite element methods using cut elements. II: A stabilized Nitsche method. (English) Zbl 1316.65099
Summary: We extend the classical Nitsche type weak boundary conditions to a fictitious domain setting. An additional penalty term, acting on the jumps of the gradients over element faces in the interface zone, is added to ensure that the conditioning of the matrix is independent of how the boundary cuts the mesh. Optimal a priori error estimates in the \(H^{1}\)- and \(L^{2}\)-norms are proved as well as an upper bound on the condition number of the system matrix.
Part I has been published in [Comput. Methods Appl. Mech. Eng. 199, No. 41–44, 2680–2686 (2010; Zbl 1231.65207)], Part III in [ESAIM, Math. Model. Numer. Anal. 48, No. 3, 859–874 (2014; Zbl 1416.65437)].

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N85 Fictitious domain methods for boundary value problems involving PDEs
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[1] Becker, R.; Burman, E.; Hansbo, P., A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity, Comput. methods appl. mech. engrg., 198, 41-44, 3352-3360, (2009) · Zbl 1230.74169
[2] Burman, E.; Hansbo, P., Fictitious domain finite element methods using cut elements: I. A stabilized Lagrange multiplier method, Comput. methods appl. mech. engrg., 199, 41-44, 2680-2686, (2010) · Zbl 1231.65207
[3] Dautray, R.; Lions, J.-L., Mathematical analysis and numerical methods for science and technology, vol. 2: functional and variational methods, (1988), Springer-Verlag Berlin
[4] Ern, A.; Guermond, J.-L., Theory and practice of finite elements, Applied mathematical sciences, vol. 159, (2004), Springer-Verlag New York, NY
[5] Hansbo, A.; Hansbo, P., An unfitted finite element method, based on nitscheʼs method, for elliptic interface problems, Comput. methods appl. mech. engrg., 191, 47-48, 5537-5552, (2002) · Zbl 1035.65125
[6] Hansbo, P., Nitscheʼs method for interface problems in computational mechanics, GAMM-mitt., 28, 2, 183-206, (2005) · Zbl 1179.65147
[7] Nitsche, J., Über ein variationsprinzip zur Lösung von Dirichlet-problemen bei verwendung von teilräumen, die keinen randbedingungen unterworfen sind, Abh. math. semin. univ. hambg., 36, 9-15, (1971) · Zbl 0229.65079
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