In mathematics, numerical analysis, and numerical partial differential equations, domain decomposition methods solve a boundary value problem by splitting it into smaller boundary value problems on subdomains and iterating to coordinate the solution between the adjacent subdomains. A corse problem with one or fiew unknows per subdomain is used to further coordinate the solution between the subdomains globally.
We consider the following laplacian boundary value problem
\begin{equation} \left \{ \begin{aligned} & -u"(x) = f(x) \quad \text{in} \quad ]0,1[ \\ & u(0) =\alpha, ~ u(1) = \beta \end{aligned} \right. \label{eq:30} \end{equation}
where \(\alpha, \beta \in \mathbb R.\)
The schwartz overlapping multiplicative algorithm with dirichlet interface conditions for this problem at \(n^{th}\) iteration is given by
\begin{equation} \label{eq:31} \left \{ \begin{aligned} -u_1"^n(x) & = f(x) \quad \text{in} \quad ]0,b[ \\ u_1^n(0) & = \alpha \\ u_1^n(b) & = u_2^{n-1}(b) \end{aligned} \right. \qquad \text{and} \qquad \left \{ \begin{aligned} -u_2"^n(x) & = f(x) \quad \text{in} \quad ]a,1[ \\ u_2^n(1) & = \beta \\ u_2^n(a) & = u_1^n(a) \end{aligned} \right. \end{equation}
where \( n \in \mathbb N^*, a, b \in \mathbb R \) and \(a < b\).
Let \(e_i^n = u_i^n-u~(i=1,2)\), the error at \(n^{th}\) iteration relative to the exact solution, the convergence rate is given by
\begin{equation} \rho = \frac{\vert e_1^n \vert}{\vert e_1^{n-1} \vert} = \frac{a}{b}\frac{1-b}{1-a} = \frac{\vert e_2^n \vert}{\vert e_2^{n-1} \vert} . \label{eq:32} \end{equation}
find \(u\) such that
\begin{equation*} \int_0^b u_1'v' = \int_0^b fv \quad \forall v \qquad \text{in the first subdomain} ~\Omega_1 = ]0,b[ \end{equation*}
\begin{equation*} \int_a^1 u_2'v' = \int_a^1 fv \quad \forall v \qquad \text{in the second subdomain} ~ \Omega_2 = ]a,1[ \end{equation*}
We consider the following laplacian boundary value problem
\begin{equation} \left \{ \begin{aligned} -\Delta u & = f \quad \text{in} \quad \Omega \\ u & = g \quad \text{on} \quad \partial\Omega \end{aligned} \right. \label{eq:33} \end{equation}
where \(\Omega \subset \mathbb R^d, d=2,3\) and \(g\) is the dirichlet boundary value.
The schwartz overlapping multiplicative algorithm with dirichlet interface conditions for this problem on two subdomains \(\Omega_1\) and \(\Omega_2\) at \(n^{th}\) iteration is given by
\begin{equation} \label{eq:34} \left \{ \begin{aligned} -\Delta u_1^n & = f \quad \qquad \text{in} \quad \Omega_1 \\ u_1^n & = g \quad \qquad \text{on} \quad \partial \Omega_1^{ext}\\ u_1^n & = u_2^{n-1} \quad ~~ \text{on} \quad \Gamma_1 \end{aligned} \right. \qquad \text{and} \qquad \left \{ \begin{aligned} - \Delta u_2^n & = f \quad \qquad \text{in} \quad \Omega_2 \\ u_2^n & = g \quad \qquad \text{on} \quad \partial \Omega_2^{ext}\\ u_2^n & = u_1^n \qquad~~ \text{on} \quad \Gamma_2 \end{aligned} \right. \end{equation}
\begin{equation*} \begin{aligned} \int_{\Omega_i} \nabla u_i \cdot \nabla v = \int_{\Omega_i} fv \quad \forall~ v,~i=1,2. \end{aligned} \end{equation*}
The numerical results presented in the following table correspond to the partition of the global domain \(\Omega\) in two subdomains \(\Omega_1\) and \(\Omega_2\) and the following configuration:
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Geometry |
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Nomber of iterations | \(\mathbf {\| u_1-u_{ex}\|_{L_2} }\) | \(\mathbf{\| u_2-u_{ex}\|_{L_2}}\) |
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11 | 2.52e-8 | 2.16e-8 |
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Isovalues of Solution in 2D |
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We consider at the continuous level the Dirichlet-to-Neumann(DtN) map on \(\Omega\), denoted by DtN \(_{\Omega}\). Let \(u: \Gamma \longmapsto \mathbb R, \)
\begin{equation*} \label{eq:35} \text{DtN}_{\Omega}(u) = \kappa \frac{\partial v}{ n} \Big |_{\Gamma} \end{equation*} where $v$ satisfies \begin{equation} \label{eq:36} \left\{ \begin{aligned} & \mathcal L(v):= (\eta - \text{div}(\kappa \nabla))v = 0 & \text{dans} \quad \Omega,\\ & v = u & \text{sur} \quad \Gamma \end{aligned} \right. \end{equation*}
where \(\Omega\) is a bounded domain of \(\mathbb R^d\) (d=2 or 3), and \(\Gamma\) it border, \(\kappa\) is a positive diffusion function which can be discontinuous, and \(\eta \geq 0\). The eigenmodes of the Dirichlet-to-Neumann operator are solutions of the following eigenvalues problem
\begin{equation} \label{eq:37} \text{DtN}_{\Omega}(u) = \lambda \kappa u \end{equation}
To obtain the discrete form of the DtN map, we consider the variational form of \((1)\). let's define the bilinear form \(a : H^1(\Omega) \times H^1(\Omega) \longrightarrow \mathbb R \),
\begin{equation*} \label{eq:41} a(w,v) := \int_\Omega \eta w v + \kappa \nabla w \cdot \nabla v . \end{equation*}
With a finite element basis \(\{ \phi_k \}\), the coefficient matrix of a Neumann boundary value problem in \(\Omega\) is
\begin{equation*} \label{eq:42} A_{kl} := \int_\Omega \eta \phi_k \phi_l + \kappa \nabla \phi_k \cdot \nabla \phi_l . \end{equation*}
A variational formulation of the flux reads
\begin{equation*} \label{eq:43} \int_\Gamma \kappa \dfrac{\partial v}{\partial n} \phi_k = \int_\Omega \eta v \phi_k + \kappa \nabla v \cdot \nabla \phi_k \quad \forall~ \phi_k. \end{equation*}
So the variational formulation of the eigenvalue problem \((2)\) reads
\begin{equation} \label{eq:40} \int_\Omega \eta v \phi_k + \kappa \nabla v \cdot \nabla \phi_k = \lambda \int_\Gamma \kappa v \phi_k \quad \forall~ \phi_k. \end{equation}
Let \(B\) be the weighted mass matrix
\begin{equation*} \label{eq:44} (B)_{kl} = \int_\Gamma \kappa \phi_k \phi_l \end{equation*}
The compact form of \((3)\) is
\begin{equation} \label{eq:45} Av = \lambda B v \end{equation}
Assembly of the right hand side \( B = \int_\Gamma \kappa v w \)
Assembly of the left hand side \( A = \int_\Omega \eta v w + \kappa \nabla v \cdot \nabla w \)
solve the eigenvalue problem \( Av = \lambda B v \)
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Three eigenmodes |
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These numerical solutions correspond to the following configuration :