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Resonant Scattering and Generation of WavesNumerical Solution of the Nonlinear Boundary Value Problem

Resonant Scattering and Generation of Waves: Numerical Solution of the Nonlinear Boundary Value... [The finite element method is based on the weak formulation (2.5). We consider N∈N,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\in \mathbb {N},$$\end{document}N≥2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 2,$$\end{document} nodes zjj=1N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ z_j\right\} _{j=1}^N$$\end{document} such that -2πδ=:z1<z2<…<zN-1<zN:=2πδ,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-2\pi \delta =:z_1<z_2<\ldots<z_{N-1}<z_N:=2\pi \delta ,$$\end{document} and define the subintervals Ij:=zj,zj+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {I}_j:=\left( z_j,z_{j+1}\right) $$\end{document} with the lengths hj:=zj+1-zj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_j:=z_{j+1}-z_j$$\end{document} and the parameter h:=maxj∈{1,…,N-1}hj.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h:=\max _{j\in \{1,\ldots ,N-1\}}h_j.$$\end{document} Then, for j∈{1,…,N}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j\in \{1,\ldots ,N\}$$\end{document} we introduce the basis functions ψj:Icl→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi _j:\;\mathscr {I}^\text {cl}\rightarrow \mathbb {R}$$\end{document} by the formula ψj(z):=z-zj-1/hj-1,z∈Ij-1andj≥2,zj+1-z/hj,z∈Ijandj≤N-1,0,otherwise\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \psi _j(z) :=\left\{ \begin{matrix} \left( z-z_{j-1}\right) /{h_{{\;j-1}}},&{} z\in \mathscr {I}_{j-1} \text{ and } j\ge 2,\\ \left( z_{j+1}-z\right) /{h_j},&{} z\in \mathscr {I}_j \text{ and } j\le N-1,\\ 0,&{} \text{ otherwise } \end{matrix}\right. $$\end{document}and the corresponding linear spaces Vh:=span{ψj}j=1N:={vh=∑j=1Nλjψj:λj∈C},Vh:=Vh3.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ V_h:=\mathop {\mathrm {span}}\{\psi _j\}_{j=1}^N :=\Big \{v_h=\sum _{j=1}^N\lambda _j\psi _j:\; \lambda _j\in \mathbb {C}\Big \}, \qquad {\mathbf V }_h:=V_h^3. $$\end{document}] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

Resonant Scattering and Generation of WavesNumerical Solution of the Nonlinear Boundary Value Problem

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Publisher
Springer International Publishing
Copyright
© Springer International Publishing AG, part of Springer Nature 2019
ISBN
978-3-319-96300-6
Pages
101 –113
DOI
10.1007/978-3-319-96301-3_5
Publisher site
See Chapter on Publisher Site

Abstract

[The finite element method is based on the weak formulation (2.5). We consider N∈N,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\in \mathbb {N},$$\end{document}N≥2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 2,$$\end{document} nodes zjj=1N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ z_j\right\} _{j=1}^N$$\end{document} such that -2πδ=:z1<z2<…<zN-1<zN:=2πδ,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-2\pi \delta =:z_1<z_2<\ldots<z_{N-1}<z_N:=2\pi \delta ,$$\end{document} and define the subintervals Ij:=zj,zj+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {I}_j:=\left( z_j,z_{j+1}\right) $$\end{document} with the lengths hj:=zj+1-zj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_j:=z_{j+1}-z_j$$\end{document} and the parameter h:=maxj∈{1,…,N-1}hj.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h:=\max _{j\in \{1,\ldots ,N-1\}}h_j.$$\end{document} Then, for j∈{1,…,N}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j\in \{1,\ldots ,N\}$$\end{document} we introduce the basis functions ψj:Icl→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi _j:\;\mathscr {I}^\text {cl}\rightarrow \mathbb {R}$$\end{document} by the formula ψj(z):=z-zj-1/hj-1,z∈Ij-1andj≥2,zj+1-z/hj,z∈Ijandj≤N-1,0,otherwise\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \psi _j(z) :=\left\{ \begin{matrix} \left( z-z_{j-1}\right) /{h_{{\;j-1}}},&{} z\in \mathscr {I}_{j-1} \text{ and } j\ge 2,\\ \left( z_{j+1}-z\right) /{h_j},&{} z\in \mathscr {I}_j \text{ and } j\le N-1,\\ 0,&{} \text{ otherwise } \end{matrix}\right. $$\end{document}and the corresponding linear spaces Vh:=span{ψj}j=1N:={vh=∑j=1Nλjψj:λj∈C},Vh:=Vh3.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ V_h:=\mathop {\mathrm {span}}\{\psi _j\}_{j=1}^N :=\Big \{v_h=\sum _{j=1}^N\lambda _j\psi _j:\; \lambda _j\in \mathbb {C}\Big \}, \qquad {\mathbf V }_h:=V_h^3. $$\end{document}]

Published: Jul 27, 2018

Keywords: Nonlinear Boundary Value Problems; Discrete Finite Element Formulation; Component-wise Application; Layer Partition; Vector Version

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