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Cut finite element methods for coupled bulk–surface problems

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Abstract

We develop a cut finite element method for a second order elliptic coupled bulk-surface model problem. We prove a priori estimates for the energy and \(L^2\) norms of the error. Using stabilization terms we show that the resulting algebraic system of equations has a similar condition number as a standard fitted finite element method. Finally, we present a numerical example illustrating the accuracy and the robustness of our approach.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University College London, WC1E 6BT, London, UK

    Erik Burman

  2. Department of Mechanical Engineering, Jönköping University, 55111, Jönköping, Sweden

    Peter Hansbo

  3. Department of Mathematics and Mathematical Statistics, Umeå University, 90187, Umeå, Sweden

    Mats G. Larson

  4. Department of Mathematics, KTH Royal Institute of Technology, 100 44, Stockholm, Sweden

    Sara Zahedi

Authors
  1. Erik Burman
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  2. Peter Hansbo
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  3. Mats G. Larson
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  4. Sara Zahedi
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Corresponding author

Correspondence to Peter Hansbo.

Additional information

This research was supported in part by EPSRC, UK, Grant No. EP/J002313/1, the Swedish Foundation for Strategic Research Grant No. AM13-0029, the Swedish Research Council Grants Nos. 2011-4992, 2013-4708, and 2014-4804, and Swedish strategic research programme eSSENCE.

Appendix

Appendix

Here we will give some details on the inequalities (4.5). First we recall that

$$\begin{aligned} q_h(x) = x + \gamma _h(x)n(x) \quad x \in \Gamma \end{aligned}$$
(6.2)

Now using the definition of the closest point mapping

$$\begin{aligned} y = p(y) + \rho (y)n^e(y) \quad y \in {\Gamma _h}\end{aligned}$$
(6.3)

Setting \(x = p(y)\) in (6.2) we have

$$\begin{aligned} y = p(y) + \gamma _h(p(y)) n^e(y)\quad y \in {\Gamma _h}\end{aligned}$$
(6.4)

and therefore, by uniqueness, \(\rho (y) = \gamma _h(p(y)), \forall y \in {\Gamma _h}\). Thus we have \(\gamma _h = \rho ^L\) and we immediately obtain the first inequality in (4.5) since

$$\begin{aligned} \Vert \gamma _h\Vert _{L^\infty (\Gamma )} = \Vert \rho ^L\Vert _{L^\infty (\Gamma )} = \Vert \rho \Vert _{L^\infty ({\Gamma _h})}\lesssim h^2 \end{aligned}$$
(6.5)

Next using (4.37) we have the identity

$$\begin{aligned} \nabla _\Gamma \gamma _h = \nabla _\Gamma \rho ^L = DF_{h,\Gamma }^T (\nabla _{\Gamma _h}\rho )^L = DF_{h,\Gamma }^T (P_{\Gamma _h}n^e)^L \end{aligned}$$
(6.6)

Estimating the right hand side using (4.24) and (4.96) we finally obtain

$$\begin{aligned} \Vert \nabla _\Gamma \gamma _h \Vert _\Gamma \lesssim \Vert \nabla _{\Gamma _h}\rho \Vert _{\Gamma _h}\lesssim \Vert P_{\Gamma _h}n^e \Vert _{\Gamma _h}\lesssim h \end{aligned}$$
(6.7)

which is the second bound in (4.5).

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Burman, E., Hansbo, P., Larson, M.G. et al. Cut finite element methods for coupled bulk–surface problems. Numer. Math. 133, 203–231 (2016). https://doi.org/10.1007/s00211-015-0744-3

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  • DOI: https://doi.org/10.1007/s00211-015-0744-3

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