Cut finite element methods for coupled bulk–surface problems

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.

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).