A Posteriori 𝑳∞(𝑳𝟐) and 𝑳∞(𝑯𝟏) Error Analysis of Semdiscrete Semilinear Parabolic Problems

Author: Younis A. Sabawi^1
1Department of Mathematics, Faculty of Science and Health, Koya University, Koya, Iraq
¹Mathematics Education Department, Faculty of Education, Tishk International University, Erbil, Iraq

Abstract: This paper aims to construct a posteriori error bounds for semilinear parabolic equations. The derivation of this bound is inspired by Makridakis and Nochetto 2003. Some challenges have been addressed through Lipschitz conditions and Gronwall’s inequality. The curtail idea for proving these estimators is to reduce the computation of schemes.

Keywords: Posteriori Error Estimate, Semilinear Parabolic Problems, Finite Element Methods

Download the PDF Document

Doi: 10.23918/eajse.v8i1p65

Published: May 30, 2022

References

Ainsworth, M., & Oden, J. T. (2000). A posteriori error estimation in finite element analysis. New York: Wiley-Interscience [John Wiley & Sons]. Retrieved from https://dx.doi.org/10.1002/9781118032824 doi: 10.1002/9781118032824

Akrivis, G., Makridakis, C., & Nochetto, R. H. (2009). Optimal order a posteriori error estimates for a class of Runge-Kutta and Galerkin methods. Numer. Math., 114(1), 133–160. Retrieved from https://dx.doi.org/10.1007/s00211-009-0254-2 doi: 10.1007/s00211-009-0254-2

Bänsch, Karakatsani, and Makridakis (2012). A posteriori error control for fully dis- crete CrankNicolson schemes. SIAM J. Numer. Anal., 50(6), 2845–2872. Retrieved from https://dx.doi.org/10.1137/110839424 doi: 10.1137/110839424

Cangiani, A., Georgoulis, E. H., & Sabawi, Y. A. (2020). Convergence of an adaptive discontinuous galerkin method for elliptic interface problems. Journal of Computational and Applied Mathematics, 367, 112397.

Cangiani, A., Georgoulis, E., & Sabawi, Y. (2018). Adaptive discontinuous galerkin methods for elliptic interface problems. Mathematics of Computation, 87(314), 2675–2707.

Demlow, A., Lakkis, O., & Makridakis, C. (2009). A posteriori error estimates in the maximum norm for parabolic problems. SIAM J. Numer. Anal., 47(3), 2157–2176. Retrieved from https://dx.doi.org/10.1137/070708792

Hussein, Y. A. (2011). Combination between single diagonal implicit and explicit runge kutta (sdimex- rk) methods for solving stiff differential equations. Tikrit Journal of Pure Science, 16(1).

Khalaf, A. D., Zeb, A., Sabawi, Y. A., Djilali, S., & Wang, X. (2021). Optimal rates for the parameter prediction of a Gaussian Vasicek process. The European Physical Journal Plus, 136(8), 1-17.

Kopteva, N., & Linss, T. (2013). Maximum norm a posteriori error estimation for parabolic problems using elliptic reconstructions. SIAM J. Numer. Anal., 51(3), 1494–1524. Retrieved from https://dx.doi.org/10.1137/110830563

Lakkis, O., & Makridakis, C. (2006). Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems. Math. Comp., 75(256), 1627–1658. Retrieved from https://dx.doi.org/10.1090/S0025-5718-06-01858-8

Makridakis, C. (2007). Space and time reconstructions in a posteriori analysis of evolution problems. In ESAIM Proceedings. Vol. 21 (2007) [Journe ́es d’Analyse Fonctionnelle et Nume ́rique en l’honneur de Michel Crouzeix] (Vol. 21, pp. 31–44). EDP Sci., Les Ulis. Retrieved from https://dx.doi.org/10.1051/proc:072104

Makridakis, C., & Nochetto, R. H. (2003). Elliptic reconstruction and a posteriori error esti- mates for parabolic problems. SIAM J. Numer. Anal., 41(4), 1585–1594. Retrieved from https://dx.doi.org/10.1137/S0036142902406314

Manaa, S. A., Moheemmeed, M. A., & Hussien, Y. A. (2010). A numerical solution for sine-gordon type system. Tikrit Journal of Pure Science, 15(3).

Pirdawood, M. A., & Sabawi, Y. A. (2021, September). High-order solution of Generalized Burgers–Fisher Equation using compact finite difference and DIRK methods. In Journal of Physics: Conference Series (Vol. 1999, No. 1, p. 012088). IOP Publishing.

Sabawi Y. A. (2017). Adaptive discontinuous galerkin methods for interface problems (Unpublished doctoral dissertation). PhD Thesis.

Sabawi, Y. A. (2019). A posteriori 𝐿∞(𝐻1) error bound in finite element approximation of semdiscrete semilinear parabolic problems. In 2019 first international conference of computer and applied sciences (cas) (pp. 102–106).

Sabawi, Y. A. (2020). A posteriori error analysis in finite element approximation for fully discrete semilinear parabolic problems. In Finite element methods and their applications.IntechOpen.

Sabawi, Y. A. (2021). A Posteriori 𝐿∞(𝐿2) + 𝐿2(𝐻1)–Error Bounds in Discontinuous Galerkin Methods For Semidiscrete Semilinear Parabolic Interface Problems. Baghdad Science Journal, 18(3).

Sabawi, Y. A. (2021). Posteriori error bound for fullydiscrete semilinear parabolic integrodifferential equations. In Journal of physics: Conference series (Vol. 1999, p. 012085).

Sabawi, Y. A., Pirdawood, M. A., & Khalaf, A. D. (2021). Semi-implicit and explicit runge kutta methods for stiff ordinary differential equations. In Journal of Physics: Conference Series (Vol. 1999, p. 012100).

Sabawi, Y. A., Pirdawood, M. A., & Rasool, H. M. (2021). Model reduction and implicit-explicit runge-kutta methods for nonlinear stiff initial-value problems, “. In Seventh international scientific conference Iraqi al khwarizmi society”, Mosul, Iraq.

Sabawi, Y. A., Pirdawood, M. A., & Sadeeq, M. I. (2021). A compact fourth-order implicit-explicit runge-kutta type method for solving diffusive lotka–volterra system. In Journal of Physics: Conference Series (Vol. 1999, p. 012103).