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Finite elements with mesh refinement for elastic wave propagation in polygons

by F. Müller and Ch. Schwab

(Report number 2014-31)

Abstract
Error estimates for the space-semidiscrete Finite Element approximation of solutions to initial boundary value problems for linear, second-order hyperbolic systems in bounded polygons \(G\subset \mathbb{R}^2\) with straight sides are presented. Using recent results on corner asymptotics of solutions of linear wave equations with time-independent coefficients in conical domains, it is shown that continuous, simplicial Lagrangian Finite Elements of uniform polynomial degree \(p\geq 1\) with either suitably graded mesh refinement or with bisection tree mesh refinement towards the corners of \(G\), achieve the (maximal) asymptotic rate of convergence \(O(N^{-p/2})\), where \(N\) denotes the number of degrees of freedom spent for the Finite Element space semidiscretization. In the present analysis, Dirichlet, Neumann and mixed boundary conditions are considered. Numerical experiments which confirm the theoretical results are presented for linear elasticity.

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

@Techreport{MS14_581,
  author = {F. M\"uller and Ch. Schwab},
  title = {Finite elements with mesh refinement for elastic wave propagation in polygons},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2014-31},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2014/2014-31.pdf },
  year = {2014}
}

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