Practical error analysis for the three-level bilinear FEM and finite-difference scheme for the 1D wave equation with non-smooth data

Alexander Zlotnik; Olga Kireeva

doi:10.3846/mma.2018.022

DOI: https://doi.org/10.3846/mma.2018.022

Abstract

We deal with the standard three-level bilinear FEM and finite-difference scheme with a weight to solve the initial-boundary value problem for the 1D wave equation. We consider the rich collection of initial data and the free term which are the Dirac δ-functions, discontinuous, continuous but with discontinuous derivatives and from the Sobolev spaces, accomplish the practical error analysis in the L², L¹, energy and uniform norms as the mesh refines and compare results with known theoretical error bounds.

Keyword : 1D wave equation, non-smooth data, bilinear FEM, finite-difference scheme, practical error analysis

How to Cite

Zlotnik, A., & Kireeva, O. (2018). Practical error analysis for the three-level bilinear FEM and finite-difference scheme for the 1D wave equation with non-smooth data. Mathematical Modelling and Analysis, 23(3), 359-378. https://doi.org/10.3846/mma.2018.022

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Jun 14, 2018

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References

[1] B.S. Jovanović and E. Süli. Analysis of finite difference schemes for linear partial differential equations with generalized solutions. Springer, London, 2014. https://doi.org/10.1007/978-1-4471-5460-0.

[2] O.A. Ladyzhenskaya. The boundary value problems of mathematical physics. Springer, Berlin, 1985. https://doi.org/10.1007/978-1-4757-4317-3.

[3] A.A. Samarskii. The theory of difference schemes. Marcel Dekker, New York-Basel, 2001. https://doi.org/10.1201/9780203908518.

[4] M. Tang. Second order all speed method for the isentropic Euler equations. Kinetic Relat. Models, 5(1):155-184, 2012. https://doi.org/10.3934/krm.2012.5.155.

[5] P. Trautmann, B. Vexler and A. Zlotnik. Finite element error analysis for measure-valued optimal control problems governed by a 1D wave equation with variable coefficients. Math. Control Relat. Fields, 8(2):411-449, 2018. https://doi.org/10.3934/mcrf.2018017.

[6] S.B. Zaitseva and A.A. Zlotnik. Exact estimates for the error gradient of locally one-dimensional methods for multidimensional equation of heat conduction. Russian Math., 41(4):51-65, 1997.

[7] A. Zlotnik and R. Čiegis. A "converse" stability condition is necessary for a compact higher order scheme on non-uniform meshes for the timedependent Schrodinger equation. Appl. Math. Letters, 80:35-40, 2018. https://doi.org/10.1016/j.aml.2018.01.005.

[8] A.A. Zlotnik. Projection-difference schemes for non-stationary problems with nonsmooth data. PhD thesis, Lomonosov Moscow State University, 1979. (in Russian)

[9] A.A. Zlotnik. Lower error estimates for three-layer difference methods of solving the wave equation with data from Hölder spaces. Math. Notes, 51(3):321-323, 1992. https://doi.org/10.1007/BF01206402.

[10] A.A. Zlotnik. Convergence rate estimates of finite-element methods for second order hyperbolic equations. In G.I. Marchuk (Ed.), Numerical methods and applications, pp. 155-220. CRC Press, Boca Raton, 1994.