Numerical Treatment of Elliptic Problems Nonlinearly Coupled Through the Interface

Abstract

This work is devoted to the study of the numerical treatment of linear elliptic problems in adjoined domains nonlinearly coupled at the interface. The problem arises in semi-discretization of mass diffusion problems typically when an osmotic effect is taken into account. Convergence of both the Conjugate Gradient and the Fixed Point method are considered and compared.

Appendix: Fixed Point Iteration

In this section we consider a fixed point procedure for problem (1.1).

Recall definition (2.1) and consider:

$$\begin{aligned}&Do\ (until\ convergence)\nonumber \\&\qquad Calculate\ \mathbf{u}^{(n)} = \mathbf{T}(g(\mathbf{u}^{(n-1)})) \end{aligned}$$

(4.1)

For our analysis we need to assume that $g(\zeta ,\xi )$ is locally lipschitz:

$$\begin{aligned} g(\zeta _1,\xi _1)- g(\zeta _2,\xi _2) \le l_K \left[ (\zeta _1-\zeta _2)^2+(\xi _1-\xi _2)^2 \right] ^{1/2}\quad \forall (\zeta _i,\xi _i)\in [0,K]^2.\quad \end{aligned}$$

(4.2)

In this hypothesis we can prove that the fixed point iterations are convergent. Our proof relies on a classic trace estimate, see ad example [20]:

Theorem 4.1

(Trace inequality) Let $\Omega$ be regular and take $v \in H^1(\Omega )$. Then $\forall \varepsilon >0$ there is $K_\varepsilon$ such that:

$$\begin{aligned} \int \limits _{\partial \Omega } v^2 \le \varepsilon \Vert \nabla v \Vert ^2_{L^2 (\Omega )} +K_\varepsilon \Vert v\Vert ^2_{L^2(\Omega )} . \end{aligned}$$

(4.3)

In order to prove convergence, take $e_i^n=u_i^{(n)}-u_i$. By Eq. (4.1), function $e_i^n$ solves the following problem:

$$\begin{aligned} a_i\left( e^n_i, v_i\right) + \alpha \left( e^n_i, v_i\right) _{\Omega _i} = \left( -1\right) ^{i+1} \left( g\left( u_1^{\left( n-1\right) },u_2^{\left( n-1\right) }\right) -g\left( u_1,u_2\right) ,v_i\right) _{\Gamma }\ . \end{aligned}$$

Now, taking in the previous equation as test function $v_i$ the same function $e_i^n$ and summing the two one obtains:

$$\begin{aligned}&a_1\left( e^n_1, e^n_1\right) + a_2\left( e^n_2, e^n_2\right) + \alpha \left( e^n_1, e^n_1\right) _{\Omega _1} + \alpha \left( e^n_2, e^n_2\right) _{\Omega _2}\nonumber \\&\quad = \left( g\left( u_1^{\left( n-1\right) },u_2^{\left( n-1\right) }\right) -g\left( u_1,u_2\right) ,e^n_1-,e^n_2\right) _{\Gamma }. \end{aligned}$$

Now we notice that $a_i(e^n_i, e^n_i) \equiv \Vert \nabla e^n_i \Vert ^2_{L^2 (\Omega _i)}$ and $\alpha (e^n_i, e^n_i)_{\Omega _i} \equiv \alpha \Vert e^n_i \Vert ^2_{L^2 (\Omega _i)}$ thus the previous can be written as:

$$\begin{aligned}&\Vert \nabla e^n_1 \Vert ^2_{L^2 (\Omega _1)} + \Vert \nabla e^n_2 \Vert ^2_{L^2 (\Omega _2)} + \alpha \left[ \Vert e^n_1 \Vert ^2_{L^2 (\Omega _1)} + \Vert e^n_2 \Vert ^2_{L^2 (\Omega _2)} \right] \nonumber \\&\quad = \left( g(u_1^{(n-1)},u_2^{(n-1)})-g(u_1,u_2),e^n_1-,e^n_2\right) _{\Gamma } . \end{aligned}$$

In order to estimate the right hand side, we use the lipschitz condition, the triangular, Jensen and Joung inequalities to obtain:

$$\begin{aligned}&\left( g(u_1^{(n-1)},u_2^{(n-1)})-g(u_1,u_2),e^n_1-,e^n_2\right) _{\Gamma }\\&= \int \limits _\Gamma \left[ g(u_1^{(n-1)},u_2^{(n-1)}) - g(u_1,u_2)\right] [e_1^n -e^n_2 ] \\&\le \int \limits _\Gamma \left| \left[ g(u_1^{(n-1)},u_2^{(n-1)}) - g(u_1,u_2)\right] [e_1^n -e^n_2 ] \right| \\&\le \int \limits _\Gamma l_K \left[ (e_1^{n-1})^2 + (e_2^{n-1})^2 \right] ^{1/2}\left| e_1^n -e^n_2 \right| \\&\le \Vert l_K \Vert _{L^1(\Gamma )} \int \limits _\Gamma \left[ (e_1^{n-1})^2 + (e_2^{n-1})^2\right] ^{1/2} \left( |e_1^n| + |e^n_2 |\right) \\&= \Vert l_K \Vert _{L^1(\Gamma )} \left[ \int \limits _\Gamma [ (e_1^{n-1})^2 + (e_2^{n-1})^2 ]^{1/2} |e_1^n| + \int \limits _\Gamma [ (e_1^{n-1})^2 +(e_2^{n-1})^2 ]^{1/2} |e^n_2 |\right] \\&\le \Vert l_K \Vert _{L^1(\Gamma )} \left[ \int \limits _\Gamma \left\{ \dfrac{1}{2} [ (e_1^{n-1})^2 + (e_2^{n-1})^2 ] + \dfrac{1}{2} |e_1^n|^2\right\} \right. \nonumber \\&\left. + \int \limits _\Gamma \left\{ \dfrac{1}{2} [ (e_1^{n-1})^2 + (e_2^{n-1})^2 ] +\dfrac{1}{2} |e_2^n|^2\right\} \right] \\&= \dfrac{\Vert l_K \Vert _{L^1(\Gamma )} }{2} \left[ 2 \Vert e_1^{n-1} \Vert _{L^2(\Gamma )}^2 + 2 \Vert e_2^{n-1} \Vert _{L^2(\Gamma )}^2 + \Vert e_1^n \Vert {L^2(\Gamma )}^2 + e_2^n \Vert _{L^2(\Gamma )}^2 \right] . \end{aligned}$$

Summarizing we get:

$$\begin{aligned}&\Vert \nabla e_1^n\Vert ^2_{L^2(\Omega ^1)} + \Vert \nabla e_2^n\Vert ^2_{L^2(\Omega ^2)} + \alpha \left[ \Vert e_1^n\Vert ^2_{L^2(\Omega ^1)} + \Vert e_2^n\Vert ^2_{L^2(\Omega ^2)} \right] \\&\le \Vert l_K \Vert _{L^1(\Gamma )} \left[ \Vert e_1^{n-1} \Vert _{L^2(\Gamma )}^2 + \Vert e_2^{n-1} \Vert _{L^2(\Gamma )}^2\right] + \dfrac{\Vert l_K \Vert _{L^1(\Gamma )}}{2} \left[ \Vert e_1^n\Vert ^2_{L^2(\Gamma )} + \Vert e_2^n\Vert ^2_{L^2(\Gamma )}\right] \end{aligned}$$

Now we can use Theorem 3.1 at $\Vert e_i^n \Vert _{L^2(\Gamma )}$ with $\alpha = \dfrac{K_\varepsilon }{\varepsilon }$ and $k_1= \dfrac{1}{\varepsilon }= \displaystyle \left\{ \dfrac{\Vert l_K \Vert _{L^1(\Gamma )}}{2}\right.$+ $\left. \gamma \Vert l_K \Vert _{L^1(\Gamma )} \right\}$, where $\gamma >1$ is a parameter.

With all these positions we have:

$$\begin{aligned}&\left\{ { \dfrac{\Vert l_K \Vert _{L^1(\Gamma )}}{2} + \gamma \Vert l_K \Vert _{L^1(\Gamma )} }\right\} \left[ \Vert e_1^n\Vert ^2_{L^2(\Gamma )} + \Vert e_2^n\Vert ^2_{L^2(\Gamma )}\right] \\&\le \alpha \left[ \Vert e_1^n\Vert ^2_{L^2(\Omega ^1)} + \Vert e_2^n\Vert ^2_{L^2(\Omega ^2)} \right] + \left( \Vert \nabla e_1^n\Vert ^2_{L^2(\Omega ^1)} + \Vert \nabla e_2^n\Vert ^2_{L^2(\Omega ^2)}\right) \end{aligned}$$

Substituting this we have:

$$\begin{aligned} \gamma \left[ \Vert e_1^n\Vert ^2_{L^2(\Gamma )} + \Vert e_2^n\Vert ^2_{L^2(\Gamma )}\right] \le \left[ \Vert e_1^{n-1} \Vert _{L^2(\Gamma )}^2 + \Vert e_2^{n-1} \Vert _{L^2(\Gamma )}^2\right] \ . \end{aligned}$$

Now we can conclude on convergence because the parameter $\gamma$ has been chosen $\gamma >1$.