On the adaptivity analysis of the wave equation

Mustafa Khirallah, Nejmeddine Chorfi, Mohamed Abdelwahed

Abstract


The purpose of this work deals with the discretization  of a  second order linear wave equation  by the implicit Euler scheme in time and by the spectral elements method in space.  We prove that the adaptivity of the time steps can be combined with the adaptivity of the spectral mesh in an optimal way. Two families of error indicators, in time and in space, are proposed. Optimal estimates are obtained.

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References


M. Abdelwahed, N. Chorfi, The spectral discretization of the second-order wave equation, An. St. Univ. Ovidius Constant. 30 (2022), no. 3, 5-20.

M. Abdelwahed, N. Chorfi, resolution of the wave equation using the spectral method, Boundary Value Problem 2022 (2022), no. 1, Article Number 15.

M. Abdelwahed, N. Chorfi, A posteriori analysis of the spectral element discretization of a non linear heat equation, Adv. Nonlinear Anal. 10 (2021), 477-490.

M. Abdelwahed, N. Chorfi, On the convergence analysis of a time dependent elliptic equation with discontinuous coefficients, Adv. Nonlinear Anal. 9 (2020), 1145-1160.

S. Adjerid, A posteriori nite element error estimation for second order hyperbolic problems, Comput. Methods Appl. Mech. Engrg. 191, (2002), 4699-4719.

M. Ainsworth, J.T. Oden, A Posteriori Error Estimation in Finite Element Analysis, J. Wiley and Sons, New York, 2000.

A. Bergam, C. Bernardi, Z. Mghazli, A posteriori analysis of the nite element discretization of some parabolic equations, Math. Comp. 74 (2005), no. 251, 1117-1138.

C. Bernardi, Y. Maday, Spectral Methods, in Handbook of Numerical Analysis V, P.G. Ciarlet and J.-L. Lions, eds., North Holland, Amsterdam, 1997, pp. 209-485.

C. Bernardi, Y. Maday, F. Rapetti, Discretisations variationnelles de problemes aux limites elliptiques, Collection Mathematiques et Application, 45, Springer-Verlag, Paris, 2004.

W. Bangerth, R. Rannacher, Finite element approximation of the acoustic wave equation: error control and mesh adaptation, East-West J. Numer. Math. 7 (1999), 263-282.

W. Bangerth, R. Rannacher, Adaptive finite element techniques for the acoustic wave equation, J. Comput. Acoust. 9 (2001), 575-591.

I. Babuska, T. Strouboulis, The Finite Element Method and Its Reliability, Oxford University Press, Oxford, 2001.

M. Bieterman, I. Babuska, The finite element method for parabolic equations. I. A posteriori error estimation, Numer. Math. 40 (1992), 339-371.

M. Bieterman, I. Babuska, The finite element method for parabolic equations. II. A posteriori error estimation and adaptive approach, Numer. Math. 40 (1982), 373-406.

C. Bernardi, E. Suli, Time and space adaptivity for the second-order wave equation, Math. Models Methods Appl. Sci. 15 (2005), 199-225.

A. Chaoui, F. Ellaggoune, A. Guezane-Lakoud, Full discretization of wave equation, Boundary Value Problems 2015 (2015), Article Number 133. DOI 10.1186/s13661-015-0396-3.

Y. Daikh, W. Chikouche, Spectral element discretization of the heat equation with variable diusion coefficient, HAL Id: hal-01143558, https://hal.archives-ouvertes.fr/hal-01143558, Apr 2015.

K. Eriksson, C. Johnson, Adaptive finite element methods for parabolic problems. I. A linear model problem, SIAM J. Numer. Anal. 28 (1991), 43-77.

K. Eriksson, C. Johnson, Adaptive finite element methods for parabolic problems. IV. Nonlinear problems, SIAM J. Numer. Anal. 32 (1995), 1729-1749.

C. Johnson, V. Thomee, An a posteriori error estimate and adaptive timestep control for a backward Euler discretization of a parabolic problem, SIAM J. Numer. Anal. 27 (1990), 277-291.

J.L. Lions, E. Magenes, Problemes aux limites non homogenes et applications, Dunod, 1968.

N.S. Papageorgiou, V.D. Radulescu, D.D. Repovs, Nonlinear analysis-theory and methods, Springer Monographs in Mathematics, Springer, 2019.

A. T. Patera, A spectral element method for fluid dynamics: Laminar flow in a channel expansion, J. Comput. Phys. 54 (1984), 468-488.

E. Suli, A posteriori error analysis and global error control for adaptive nite volume approximations of hyperbolic problems, Numerical Analysis 1995 (Dundee 1995), 169-190, Pitman Res. Notes Math. Ser. 344. Longman, Harlow, 1996.

E. Suli, A posteriori error analysis and adaptivity for finite element approximations of hyperbolic problems, In: D. Kroner, M. Ohlberger and C. Rohde (Eds.) An Introduction to Recent Developments in Theory and Numerics for Conservation Laws. Lecture Notes in Computational Science and Engineering Volume 5, 123 -194, Springer-Verlag, 1998.

K.P. Jin, L. Wang, Uniform decay estimates for the semi-linear wave equation with locally distributed mixed-type damping via arbitrary local viscoelastic versus frictional dissipative effects, Adv. Nonlinear Anal. 12 (2023). https://doi.org/10.1515/anona-2022-0285.

R. Verfurth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Renement Techniques, Wiley et Teubner, 1996.

R. Verfurth, A posteriori error estimation techniques for non-linear elliptic and parabolic pdes, Revue europeenne des element finis 9 (2000), 377-402.

Y. Yang, B.F Zhong, On a strongly damped semilinear wave equation with time-varying source and singular dissipation, Adv. Nonlinear Anal. 12 (2023). https://doi.org/10.1515/anona-2022-0267.

J. Zhang, W. Zhang, V.D. Radulescu, Double phase problems with competing potentials concentration and multiplication of ground states, Math. Z. 301 (2022), 4037-4078.

W. Zhang, J. Zhang, V.D. Radulescu, Concentrating solutions for singularly perturbed double phase problems with nonlocal reaction, J. Differ. Equ. 347 (2023), 56-103.




DOI: https://doi.org/10.52846/ami.v50i2.1831