ON FINITE ELEMENT METHOD FOR LINEAR HYPERBOLIC INTERFACE PROBLEMS

MATTHEW O. ADEWOLE

Abstract


We investigate the error contributed by semi discretization
to the finite element solution of linear hyperbolic interface problems. With low regularity assumption on the solution across the interface, almost optimal convergence rates in L2(Ω) and H1(Ω) norms are obtained. We do not assume that the interface could be fitted exactly. Numerical experiments are presented to support the theoretical results.

Full Text:

PDF

References


R. A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace

Jovanovich, Publishers], New York-London, 1975.

M. O. Adewole, Almost optimal convergence of FEM-FDM for a linear parabolic interface problem, Electron. Trans. Numer. Anal. 46 337–358, 2017.

M. O. Adewole and V. F. Payne, Convergence of a Finite Element Solution for a Nonlinear Parabolic Equation with Discontinuous Coefficient, To appear in the Proceeding of International Conference on Contemporary Mathematics and the Real World, 2017.

I. Babuˇska, The finite element method for elliptic equations with discontinuous coefficients, Computing (Arch. Elektron. Rechnen) 5 207–213, 1970.

G. A. Baker, Error estimates for finite element methods for second order hyperbolic equations, SIAM J. Numer. Anal. 13 564–576, 1976.

G. A. Baker and J. H. Bramble, Semidiscrete and single step fully discrete approximations for second order hyperbolic equations, RAIRO Analyse Num´erique 13 75–100, 1979.

G. A. Baker and V. A. Dougalis, On the L∞-convergence of Galerkin approximations for second-order hyperbolic equations, Math. Comp. 34 401–424, 1980.

L. M. Brekhovskikh, Waves in Layered Media, Academic Press, New York, 1973.

Z. Chen and J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math. 79 175–202, 1998.

B. Deka, Finite element methods with numerical quadrature for elliptic problems with smooth interfaces, J. Comput. Appl. Math. 234 605–612, 2010.

B. Deka and T. Ahmed, Convergence of finite element method for linear second order wave equations with discontinuous coefficients, Numer. Methods Partial Differential Equations 29 (5) 1522–1542, 2013.

B. Deka and R. K. Sinha, Finite element methods for second order linear hyperbolic interface problems, Appl. Math. Comput. 218 10922–10933, 2012.

T. Dupont, L2-estimates for Galerkin methods for second order hyperbolic equations, SIAM J. Numer. Anal. 10 880–889, 1973.

L. C. Evans, Partial differential equations, Vol 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1998.

E. H. Georgoulis, O. Lakkis and C. Makridakis,A posteriori L∞(L2)-error bounds for finite element approximations to the wave equation, IMA J. Numer. Anal. 33 1245–1264, 2013.

F. Hecht, New development in freefem++, J. Numer. Math. 20 251–265, 2012.

S. Larsson and V. Thom´ee, Partial differential equations with numerical methods, vol 45 of Texts in Applied Mathematics, Springer-Verlag, Berlin, 2003.

V. F. Payne and M. O. Adewole, Error Estimates for a nonlinear parabolic interface problem on a convex polygonal domain, To appear in International Journal of Mathematical Analysis (IJMA).

J. Rauch, On convergence of the finite element method for the wave equation, SIAM J. Numer. Anal. 22 245–249, 1985.

J. Sen Gupta, R. K. Sinha, G. M. M. Reddy and J. Jain, A posteriori error analysis of two-step backward differentiation formula finite element approximation for parabolic interface problems, J. Sci. Comput. 69 406–429, 2016.

J. Sen Gupta, R. K. Sinha, G. M. M. Reddy and J. Jain, New interpolation error estimates and a posteriori error analysis for linear parabolic interface problems, Numer. Methods Partial Differential Equations 33 570–598, 2017.

R. K. Sinha and B. Deka, An unfitted finite-element method for elliptic and parabolic interface problems, IMA J. Numer. Anal. 27 529–549, 2007.

L. Song and C. Yang, Convergence of a second-order linearized BDF-IPDG for nonlinear parabolic equations with discontinuous coefficients, J. Sci. Comput. 70 662–685, 2017.

C. Yang, Convergence of a linearized second-order BDF-FEM for nonlinear parabolic interface problems, Comput. Math. Appl. 70 265–281, 2015.


Refbacks

  • There are currently no refbacks.


Copyright (c) 2018 Journal of the Nigerian Mathematical Society

Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.