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Theoretical and Numerical Investigation of the Finite Cell Method

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Abstract

We present a detailed analysis of the convergence properties of the finite cell method which is a fictitious domain approach based on high order finite elements. It is proved that exponential type of convergence can be obtained by the finite cell method for Laplace and Lamé problems in one, two as well as three dimensions. Several numerical examples in one and two dimensions including a well-known benchmark problem from linear elasticity confirm the results of the mathematical analysis of the finite cell method.

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Notes

  1. The operator of extension by \(0\) from \(V'(\Omega )\) into \(H^{-1}(\mathcal {D})\) has to be understood as the dual of the restriction operator from \(H^1_0(\mathcal {D})\) onto \(V(\Omega )\).

  2. Let \({\hat{I}}=(-1,1)\) be the reference segment. Tensor \(H^1\)-norms are defined as follows in dimension \(n=2\) and \(n=3\):

    $$\begin{aligned} \Vert u\Vert \left. \right. _{H^{1,1}({\hat{I}}^2)}^2 \!=\! \sum _{\alpha _1=0}^1\sum _{\alpha _2=0}^1 \Vert \partial _{x_1}^{\alpha _1}\partial _{x_2}^{\alpha _2} u\Vert \left. \right. _{L^2({\hat{I}}^2)}^2 \quad \text{ and }\quad \Vert u\Vert \left. \right. _{H^{1,1,1}({\hat{I}}^3)}^2 = \sum _{\alpha _1=0}^1\sum _{\alpha _2=0}^1\sum _{\alpha _3=0}^1 \Vert \partial _{x_1}^{\alpha _1}\partial _{x_2}^{\alpha _2}\partial _{x_3}^{\alpha _3} u\Vert \left. \right. _{L^2({\hat{I}}^3)}^2. \end{aligned}$$

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Authors and Affiliations

  1. Université de Rennes 1, Rennes, France

    Monique Dauge

  2. Technische Universität Hamburg-Harburg, Hamburg, Germany

    Alexander Düster

  3. Technische Universität München, Munich, Germany

    Ernst Rank

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  1. Monique Dauge
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  2. Alexander Düster
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Correspondence to Monique Dauge.

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Dauge, M., Düster, A. & Rank, E. Theoretical and Numerical Investigation of the Finite Cell Method. J Sci Comput 65, 1039–1064 (2015). https://doi.org/10.1007/s10915-015-9997-3

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