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

    Alexander Zlotnik Info
    Olga Kireeva Info

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.

Keywords:

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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June 14, 2018
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References

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.

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

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

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.

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.

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.

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.

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

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.

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.

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2018-06-14

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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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