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Finite elements with mesh refinement for elastic wave propagation in polygons
Author(s) -
Müller Fabian,
Schwab Christoph
Publication year - 2015
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.3355
Subject(s) - mathematics , finite element method , degree of a polynomial , mathematical analysis , linear elasticity , boundary value problem , bounded function , conical surface , rate of convergence , geometry , polynomial , engineering , physics , thermodynamics , channel (broadcasting) , electrical engineering
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 ⊂ R 2with 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 ≥1, with either suitably graded mesh refinement or with bisection‐tree mesh refinement toward 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 that confirm the theoretical results are presented for linear elasticity. Copyright © 2015 John Wiley & Sons, Ltd.