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Finite Elements with mesh refinement for wave equations in polygons

by F. Müller and Ch. Schwab

(Report number 2013-11)

Abstract
Error estimates for the space-semidiscrete approximation of solutions of the wave equation in polygons \(G\subset \mathbb{R}^2\) are presented. Based on corner asymptotics of solutions of the wave equation, it is shown that for continuous, simplicial Lagrangian Finite Elements of polynomial degree \(p\geq 1\) with either suitably graded mesh refinement or with bisection tree mesh refinement towards the corners of \(G\), the maximal rate of convergence \(O(N^{-p/2})\) which is afforded by the Lagrangian Finite Element approximations on quasiuniform meshes for smooth solutions is restored. Dirichlet, Neumann and mixed boundary conditions are considered. Numerical experiments which confirm the theoretical results are presented. Generalizations to nonhomogeneous coefficients and elasticity and electromagnetics are indicated.

Keywords: High order Finite Elements, Wave equation, Regularity, Weighted Sobolev spaces, Method of lines, Local mesh refinement, Graded meshes, Newest vertex bisection

@Techreport{MS13_507,
  author = {F. M\"uller and Ch. Schwab},
  title = {Finite Elements with mesh refinement for wave equations in polygons},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2013-11},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2013/2013-11.pdf },
  year = {2013}
}

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