A Unified Integral Equation Scheme for Doubly Periodic Laplace and Stokes Boundary Value Problems in Two Dimensions

Neumann-Laplace Case

Consider heat/electrical problem. A single inclusion \(\Omega\) per unit cell. \(e_1\) and \(e_2\) are the basis of the lattice on \(\mathbb{R}^2\). We working with unit square \(e_1 = (1,0)\) and \(e_2 = (0,1)\) for simplicity. Let

\[ \Omega_\Lambda :=\left\{x\in \mathbb{R}^2: x+m_1 e_1 + m_2 e_2 \in \Omega for some m_1, m_2 \in \mathbb{Z}\right\} \]

The scalar \(u\) satisfies the following BVP:

\[ \begin{aligned} \Delta u =0\quad &\text{ in } \mathbb{R}^2 \backslash \overline{\Omega_\Lambda}, \\ u_n =0 \quad &\text{ on } \partial \Omega_\Lambda, \\ u\left(x+e_{1}\right) -u(x)=p_1 \quad &\text{for all }x \in \mathbb{R}^2 \backslash \overline{\Omega_\Lambda}, \\ u(x+e_{2}) -u(x)=p_2 \quad &\text{for all }x \in \mathbb{R}^2 \backslash \overline{\Omega_\Lambda}. \end{aligned} \]

i.e. \(u\) is harmonic, has zero flux on the boundary of the inclusion, and is periodic up to a given pair of constants \(p=(p_1, p_2)\). Here \(u_n=\frac{\partial u}{\partial n}=n\cdot \nabla u\), where \(n\) is the unit normal on the relevant evaluation curve.

2.1

For each \((p_1, p_2)\) the solution to the above BVP is unique up to an additive constant.

proof

Let \(\mathcal{B}\) be any unit cell containing \(\Omega\). Use Green's first identity to get

\[ 0=\int_{\mathcal{B} \backslash \overline{\Omega}} u \Delta u =\int_{\partial \mathcal{B}} u u_{n}-\int_{\partial \Omega} u u_{n} -\int_{\mathcal{B} \backslash \overline{\Omega}}|\nabla u|^{2} \]

The first two integrals vanish by the boundary conditions, so \(\nabla u=0\) in \(\mathcal{B} \backslash \overline{\Omega}\), and thus \(u\) is constant.

To solve the BVP, we re-express it on single unit cell \(\mathcal{B}\) with coupled boundary values on the four walls comprising its boundary \(\partial \mathcal{B}:=L\cup R\cup D\cup U\). Use notation \(u_L\) to mean the restriction of \(u\) to \(L\), and so on, and \(u_{nL}\) for its normal derivative using the normal on \(L\). Consider the following BVP:

\[ \begin{aligned} \Delta u =0 \quad &\text{ in } \mathcal{B} \backslash \overline{\Omega}, \\ u_n =0 \quad &\text{ on } \partial \Omega, \\ u_R-u_L=p_1,\\ u_{nR}-u_{nL}=0,\\ u_U-u_D=p_2,\\ u_{nU}-u_{nD}=0. \end{aligned} \]

Clearly the two BVPs are equivalent. We define discrepancy of a solution \(u\) as the stack of the four functions on the left-hand side of last four equations.

The Empty Unit Cell Discrepancy BVP and Its Numerical Solution

A subproblem called "empty unit cell BVP". We seek a harmonic function \(v\) matching a given discrepancy \(g=[g_1; g_2; g_3; g_4]\)(i.e. the right-hand side of the last four equations above), That is:

\[ \begin{aligned} \Delta v =0 \quad &\text{ in } \mathcal{B}, \\ v_R-v_L=g_1,\\ v_{nR}-v_{nL}=g_2,\\ v_U-v_D=g_3,\\ v_{nU}-v_{nD}=g_4. \end{aligned} \]

2.2

A solution \(v\) exist iff \(\int_L g_2 ds+\int_D g_4 ds=0\) and is then unique up to an constant.

proof

The zero-flux condition \(\int_{\partial \mathcal{B}} v_n =0\) holds for harmonic functions. Writing \(\partial \mathcal{B}\) as the union of the four walls which gives the condition on \(g\). Uniqueness follows from the same argument as in Prop 2.1.

We now describe a numerical solution is accurate for a certain class of data \(g\) which is sufficient for our periodic scheme.

Recall the fundamental solution of the Laplace equation in 2D:

\[ G(x,y)=\frac{1}{2\pi}\log \frac{1}{r}, \quad r=||x-y|| \]