Numerical Study Of An Anisotropic Error Estimator In The L-2(H-1) Norm For The Finite Element Discretization Of The Wave Equation

Marco Picasso
2010
Article

Résumé

An anisotropic a posteriori error estimate is derived for a finite element discretization of the wave equation in two space dimensions. Only the error due to space discretization is considered, and the error estimates are derived in the nonnatural L-2(0, T; H-1(Omega)) norm using elliptic reconstruction. A numerical study of the effectivity index on unstructured, nonadapted, anisotropic meshes confirms the sharpness of the error estimator, provided the error due to time discretization is negligible compared to the finite element error. An anisotropic, adaptive finite element algorithm is then presented to control the finite element error in the L-2(0, T; H-1(Omega)) norm. Numerical results on adapted meshes indicate that the error estimator slightly underestimates the true error. We conjecture that the missing information corresponds to the interpolation error between successive meshes. It is observed that the error estimator becomes sharp again when considering the damped wave equation

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https://infoscience.epfl.ch/record/170245?ln=fr

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In this paper we derive two a posteriori upper bounds for the heat equation. A continuous, piecewise linear finite element discretization in space and the Crank-Nicolson method for the time discretization are used. The error due to the space discretization is derived using anisotropic interpolation estimates and a postprocessing procedure. The error due to the time discretization is obtained using two different continuous, piecewise quadratic time reconstructions. The first reconstruction is developed following G. Akrivis, C. Makridakis, and R. H. Nochetto [Math. Comp., 75 (2006), pp. 511-531], while the second one is new. Moreover, in the case of isotropic meshes only, upper and lower bounds are provided as in [R. Verfurth, Calcolo, 40 (2003), pp. 195 212]. An adaptive algorithm is developed. Numerical studies are reported for several test cases and show that the second error estimator is more efficient than the first one. In particular, the second error indicator is of optimal order with respect to both the mesh size and the time step when using our adaptive algorithm.

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Anisotropic Error Estimates And Space Adaptivity For A Semidiscrete Finite Element Approximation Of The Transient Transport Equation

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A stabilized semidiscrete finite element discretization of the transient transport equation is studied in the framework of anisotropic meshes. A priori and a posteriori error estimates are derived, the involved constants being independent of the mesh aspect ratio, only space discretization being considered. Numerical results on nonadapted, anisotropic meshes and small time steps confirm the sharpness of the theoretical predictions. An anisotropic, adaptive finite element algorithm is then proposed with the goal to control the L-2 error in space at final time, the time step being kept constant. Numerical results are then presented on anisotropic, adapted meshes and small time steps. Three different methods are proposed to interpolate the solution between two adapted meshes. Numerical results indicate that both the conservative interpolation method of [F. Alauzet and M. Mehrenberger, Internat. J. Numer. Methods Engrg., 84 (2010), pp. 1552-1588] or the L-2 projection advocated in [P. E. Farrell et al., Comput. Methods Appl. Mech. Engrg., 198 (2009), pp. 2632-2642] yield accurate results.

Siam Publications2013

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A posteriori error estimates for the heat equation in two space dimensions are presented. A classical discretization is used, Euler backward in time, and continuous, piecewise linear triangular finite elements in space. The error is bounded above and below by an explicit error estimator based on the residual. Numerical results are presented for uniform triangulations and constant time steps. The quality of our error estimator is discussed. An adaptive algorithm is then proposed. Successive Delaunay triangulations are generated, so that the estimated relative error is close to a preset tolerance. Again, numerical results demonstrate the efficiency of our approach. (C) 1998 Elsevier Science S.A. All rights reserved.

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