diff --git a/docs/guide/chunkgraphbvps.rst b/docs/guide/chunkgraphbvps.rst index 76a7a47..d49dc6a 100644 --- a/docs/guide/chunkgraphbvps.rst +++ b/docs/guide/chunkgraphbvps.rst @@ -69,12 +69,12 @@ the following PDE: .. math:: - \begin{align*} + \begin{aligned} (\Delta + k^2) u^{\textrm{scat}} &= 0 & \textrm{ in } \mathbb{R}^2 \setminus \Omega \; , \\ \frac{\partial u^{\textrm{scat}}}{\partial n} &= -\frac{\partial u^{\textrm{inc}}}{\partial n} & \textrm{ on } \Gamma \; , \\ \sqrt{|x|} \left( \frac{x}{|x|} \cdot \nabla u^{\textrm{scat}} - ik u^{\textrm{scat}} \right ) &\to 0 & \textrm{ as } |x|\to \infty \; , - \end{align*} + \end{aligned} The Green function for the Helmholtz equation is @@ -87,10 +87,10 @@ single and double layer potential operators .. math:: - \begin{align*} + \begin{aligned} [S_{k}\sigma](x) &:= \int_\Gamma G_k(x,y) \sigma(y) ds(y) \\ [D_{k}\sigma](x) &:= \int_\Gamma n(y)\cdot \nabla_y G_k(x,y) \sigma(y) ds(y) - \end{align*} + \end{aligned} A robust choice for the layer potential representation for this problem is a *right preconditioned combined field* layer potential, which is a linear combination @@ -112,10 +112,10 @@ results in the equation .. math:: - \begin{align*} + \begin{aligned} -\frac{\partial u^{\textrm{inc}}(x_0)}{\partial n} &= \lim_{x\in \mathbb{R}^2\setminus \Omega, x\to x_0} \beta[(S_{k} + i\alpha D_{k} S_{ik})\sigma](x) \\ &= \sigma(x_0) + \beta[S_{k}'\sigma ](x_0) + i\beta \alpha [(D_{k}' - D_{ik}')S_{ik}\sigma](x_{0}) + i \beta \alpha [S_{ik}'S_{ik}'\sigma](x_0) \;, - \end{align*} + \end{aligned} where $S_{k}'$ and $D_{k}'$ represent the normal derivatives of the single and double layer potentials respectively, and we have used @@ -186,11 +186,11 @@ for the potential $u$ on a domain $\Omega$ with boundary $\Gamma$, .. math:: - \begin{align*} + \begin{aligned} \Delta u &= 0 & \textrm{ in } \Omega \; , \\ u &= f_{D} & \textrm{ on } \Gamma_{D} \; , \\ \frac{\partial u}{\partial n} &= f_{N} & \textrm{ on } \Gamma_{N} \; , - \end{align*} + \end{aligned} where $\Gamma_{D}$ and $\Gamma_{N}$ are a partition of the boundary, i.e. $\Gamma = \Gamma_{D} \cup \Gamma_{N}$, with @@ -223,20 +223,20 @@ where .. math:: - \begin{align*} + \begin{aligned} [S_{\Gamma_{N}}\sigma_{N}](x) = \int_{\Gamma_{N}} G_0(x,y) \sigma_{N}(y) ds(y) \; ,\\ [D_{\Gamma_{D}}\sigma_{D}](x) = \int_{\Gamma_{D}} n(y) \cdot \nabla_{y} G_0(x,y) \sigma_{D}(y) ds(y) \; . - \end{align*} + \end{aligned} On imposing the boundary conditions, we get the following system of integral equations for $\sigma_{D}$ and $\sigma_{N}$ .. math:: - \begin{align*} + \begin{aligned} \sigma_{D}(x) - 2[D_{\Gamma_{D}}\sigma_{D}](x) + 2[S_{\Gamma_{N}}\sigma_{N}](x) &= f_{D} \,, \,\, \textrm{ on } \Gamma_{D} \,, \\ \sigma_{N}(x) - 2[D'_{\Gamma_{D}}\sigma_{D}](x) + 2[S'_{\Gamma_{N}}\sigma_{N}](x) &= f_{N} \,,\,\, \textrm{ on } \Gamma_{N} \,, - \end{align*} + \end{aligned} where $S'$ and $D'$ are restrictions of the normal derivatives of $S$ and $D$ respectively. diff --git a/docs/guide/simplebvps.rst b/docs/guide/simplebvps.rst index 838058d..1d31659 100644 --- a/docs/guide/simplebvps.rst +++ b/docs/guide/simplebvps.rst @@ -26,10 +26,10 @@ value problem .. math:: - \begin{align*} + \begin{aligned} \mathcal{L} u &= 0 & \textrm{ in } \Omega \; ,\\ \mathcal{B} u &= f & \textrm{ on } \Gamma \; , - \end{align*} + \end{aligned} where :math:`\mathcal{L}` is a linear, constant coefficient PDE operator and :math:`\mathcal{B}` is a linear trace operator that @@ -87,10 +87,10 @@ Consider the Laplace Neumann boundary value problem: .. math:: - \begin{align*} + \begin{aligned} \Delta u &= 0 & \textrm{ in } \Omega \; ,\\ \frac{\partial u}{\partial n} &= f & \textrm{ on } \Gamma \; , - \end{align*} + \end{aligned} where :math:`n` is the outward normal. This is a model for the equilibrium heat distribution in a @@ -133,14 +133,14 @@ results in the equation .. math:: - \begin{align*} + \begin{aligned} f(x_0) &= \lim_{x\in \Omega, x\to x_0} n(x_0)\cdot \nabla_x \int_\Gamma G_0(x,y) \sigma(y) \, ds(y) \\ &= \frac{1}{2} \sigma(x_0) + P.V. \int_\Gamma n(x_0) \cdot \nabla_x G_0(x_0,y) \sigma(y) \, ds(y) \\ &=: \left [\left ( \frac{1}{2} \mathcal{I} + \mathcal{S}' \right ) \sigma \right ] (x_0) \; , - \end{align*} + \end{aligned} where :math:`P.V.` indicates that the integral should be interpreted in the principal value sense, :math:`\mathcal{I}` is the identity operator, and @@ -245,12 +245,12 @@ in the exterior of the object: .. math:: - \begin{align*} + \begin{aligned} (\Delta + k^2) u^{\textrm{scat}} &= 0 & \textrm{ in } \mathbb{R}^2 \setminus \Omega \; , \\ u^{\textrm{scat}} &= -u^{\textrm{inc}} & \textrm{ on } \Gamma \; , \\ \sqrt{|x|} \left( \frac{x}{|x|} \cdot \nabla u^{\textrm{scat}} - ik u^{\textrm{scat}} \right ) &\to 0 & \textrm{ as } |x|\to \infty \; , - \end{align*} + \end{aligned} The Green function for the Helmholtz equation is @@ -263,10 +263,10 @@ single and double layer potential operators .. math:: - \begin{align*} + \begin{aligned} [S\sigma](x) &:= \int_\Gamma G_k(x,y) \sigma(y) ds(y) \\ [D\sigma](x) &:= \int_\Gamma n(y)\cdot \nabla_y G_k(x,y) \sigma(y) ds(y) - \end{align*} + \end{aligned} A robust choice for the layer potential representation for this problem is a *combined field* layer potential, which is a linear combination @@ -281,10 +281,10 @@ results in the equation .. math:: - \begin{align*} + \begin{aligned} -u^{\textrm{inc}}(x_0) &= \lim_{x\in \mathbb{R}^2\setminus \Omega, x\to x_0} [(D-ik S)\sigma](x) \\ &= \frac{1}{2} \sigma(x_0) + [ (\mathcal{D} -ik \mathcal{S})\sigma ](x_0) - \end{align*} + \end{aligned} where we have used the exterior jump condition for the double layer potential. As above, the integrals in the operators restricted to the boundary must sometimes diff --git a/docs/guide/transmission.rst b/docs/guide/transmission.rst index 8980612..d3b8077 100644 --- a/docs/guide/transmission.rst +++ b/docs/guide/transmission.rst @@ -44,13 +44,13 @@ the following transmission boundary value problem .. math:: - \begin{align*} + \begin{aligned} \Delta u_{j} + k_{j}^2 u_{j} &=0 & \textrm{ in } \Omega_{j} \; , j=1,2\ldots n_{r}\\ u_{r_{\ell,+}} - u_{r_{\ell,-}} &= f_{\ell} & \textrm{ on } \Gamma_{\ell} \;, \ell=1,2,\ldots n_{e}\,, \\ \beta_{r_{\ell,+}}\frac{\partial u_{r_{\ell,+}}}{\partial n} - \beta_{r_{\ell,-}}\frac{\partial u_{r_{\ell,-}}}{\partial n} &= g_{\ell} & \textrm{ on } \Gamma_{\ell}\;, \ell=1,2,\ldots n_{e}\,,\\ \sqrt{|x|} \left( \frac{x}{|x|} \cdot \nabla u_{1} - ik_{1} u_{1} \right ) &\to 0 & \textrm{ as } |x|\to \infty \;. - \end{align*} + \end{aligned} Here $\beta_{j} = 1$ for the transverse magnetic mode and $\beta_{j} = 1/\varepsilon_{j}$ for the transverse electric mode. @@ -73,10 +73,10 @@ Recall that the single and and double layer potentials are given by .. math:: - \begin{align*} + \begin{aligned} [S_{k}\sigma](x) &:= \int_\Gamma G_k(x,y) \sigma(y) ds(y)\;, \\ [D_{k}\sigma](x) &:= \int_\Gamma n(y)\cdot \nabla_y G_k(x,y) \sigma(y) ds(y) \;, - \end{align*} + \end{aligned} where $\Gamma = \cup_{\ell=1}^{n_{e}} \Gamma_{\ell}$. @@ -85,16 +85,16 @@ and double layer potential operators as: .. math:: - \begin{align*} + \begin{aligned} u_{j}(x) &= \frac{1}{\beta_{j}}[D_{k_{j}}\sigma](x) - [S_{k_{j}}\mu](x) & \textrm{ in } \Omega_{j}\;, - \end{align*} + \end{aligned} Then, imposing the boundary conditions on this representation results in the following equation for $\sigma, \mu$: .. math:: - \begin{align*} + \begin{aligned} \begin{bmatrix} \sigma\\ \mu @@ -113,7 +113,7 @@ in the following equation for $\sigma, \mu$: \beta_{r_{\ell,+}} \beta_{r_{\ell,-}}f_{\ell}\\ g_{\ell} \end{bmatrix} \;\; \textrm{ on } \Gamma_{\ell}\;, - \end{align*} + \end{aligned} for $\ell=1,2,\ldots n_{e}$ where $\gamma_{\ell} = \frac{2}{\beta_{r_{\ell,+}} + \beta_{r_{\ell,-}}}$.