-\end{align*}\]</p><h2 id="Variational-formulation"><a class="docs-heading-anchor" href="#Variational-formulation">Variational formulation</a><a id="Variational-formulation-1"></a><a class="docs-heading-anchor-permalink" href="#Variational-formulation" title="Permalink"></a></h2><p>In this section, we present the variational formulation of the boundary value problem: Find <span>$v \in H_{per}^{1}(\Omega_{0})$</span> satisfying</p><p class="math-container">\[\int_{\Omega_{0}} \nabla v \cdot \nabla \bar{\phi} - 2i\alpha \partial_{1} v \bar{\phi} - (k^{2}q(x_{1}, x_{2}) - \alpha^{2}) v \bar{\phi} dx = 0.\]</p><p>After finite element discretization, we can obtain a generalized linear eigenvalue problem</p><p class="math-container">\[\mathbf{A}_{\alpha} \mathbf{v} = k^{2} \mathbf{B} \mathbf{v},\]</p><p>where <span>$\mathbf{A}_{\alpha}$</span> comes from</p><p class="math-container">\[\int_{\Omega_{0}} \nabla v \cdot \nabla \bar{\phi} - 2i\alpha \partial_{1} v \bar{\phi} + \alpha^{2} v \bar{\phi} dx \]</p><p>and <span>$\mathbf{B}$</span> comes from</p><p class="math-container">\[\int_{\Omega_{0}} q(x_{1}, x_{2}) v \bar{\phi} dx.\]</p><div class="admonition is-info" id="important-steps-in-the-computation-21d31f66c14e893a"><header class="admonition-header">important steps in the computation<a class="admonition-anchor" href="#important-steps-in-the-computation-21d31f66c14e893a" title="Permalink"></a></header><div class="admonition-body"><ul><li>We use the Finite element method to discretize the variational formulation. In ClosedWaveguideDispersion.jl, all Finite element codes are implemented by <a href="https://github.com/Ferrite-FEM/Ferrite.jl">Ferrite.jl</a></li><li>After the Finite element discretization, we obtain a generalized linear eigenvalue problem parametered by <span>$\alpha$</span>. We utilize <a href="https://github.com/JuliaLinearAlgebra/Arpack.jl">Arpack.jl</a> to solve the generalized linear eigenvalue problems with fixed <span>$\alpha$</span>.</li></ul></div></div></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Home</a><a class="docs-footer-nextpage" href="../homogeneous/">Homogeneous case with Neumann boundary condition »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.11.4 on <span class="colophon-date" title="Monday 19 May 2025 22:04">Monday 19 May 2025</span>. Using Julia version 1.11.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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