A Posteriori L_∞ (L_2 )+L_2 (H^1 )–Error Bounds in Discontinuous Galerkin Methods For Semidiscrete Semilinear Parabolic Interface Problems
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Abstract
The aim of this paper is to derive a posteriori error estimates for semilinear parabolic interface problems. More specifically, optimal order a posteriori error analysis in the - norm for semidiscrete semilinear parabolic interface problems is derived by using elliptic reconstruction technique introduced by Makridakis and Nochetto in (2003). A key idea for this technique is the use of error estimators derived for elliptic interface problems to obtain parabolic estimators that are of optimal order in space and time.
Received 10/7/2019, Accepted 8/6/2020, Published Online First 21/2/2021
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Adjerid S, Flaherty JE, Babuška I. A posteriori error estimation for the finite element method-of-lines solution of parabolic problems. Mathematical Models and Methods in Applied Science.1999 Mar; 9(02):261-86.
Akrivis G, Makridakis C, Nochetto RH. Optimal order a posteriori error estimates for a class of Runge–Kutta and Galerkin methods. Numerische Mathematik. 2009 Nov 1;114(1):133.
Bansch E, Karakatsani F, Makridakis C. A posteriori error control for fully discrete Crank--Nicolson schemes. SIAM Journal on numerical analysis. 2012;50(6):2845-72.
Cangiani A, Georgoulis EH, Giani S, Metcalfe S. hp-adaptive discontinuous Galerkin methods for non-stationary convection–diffusion problems. Computers and Mathematics with Applications. 2019 Nov 1; 78(9):3090-3104.
Carstensen C, Liu W, Yan N. A posteriori error estimates for finite element approximation of parabolic p-Laplacian. SIAM Journal on numerical analysis. 2006; 43(6):2294–2319
Demlow A, Lakkis O, Makridakis C. A posteriori error estimates in the maximum norm for parabolic problems. SIAM journal on numerical analysis. 2009; 47(3):2157-76.
Georgoulis EH, Lakkis O, Virtanen JM. A posteriori error control for discontinuous Galerkin methods for parabolic problems. SIAM Journal on numerical analysis. 2011; 49(2):427-58.
Kopteva N, Linss T. Maximum norm a posteriori error estimation for parabolic problems using elliptic reconstructions. SIAM Journal on numerical analysis. 2013; 51(3):1494–1524.
Kyza I, Metcalfe S, Wihler TP. hp-Adaptive Galerkin time stepping methods for nonlinear initial value problems. Journal of Scientific Computing. 2018 Apr 1; 75(1):111-2711.
Sutton O J. Long-time L^∞ (L_2 ) a posteriori error estimates for fully discrete parabolic problems. IMA Journal of Numerical analysis, 2020; 40(1):498-529.
Makridakis C. Space and time reconstructions in a posteriori analysis of evolution problems. ESAIM Proceeding, 2007; (21):31–44.
Makridakis C, Nochetto RH. Elliptic reconstruction and a posteriori error estimates for parabolic problems. SIAM Journal on numerical analysis. 2013; 41(4):1585–1594.
Makridakis C, Nochetto RH. Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems. Mathematic of Computation. 2006; 75(256):1627–1658.
Cangiani A, Georgoulis EH, Kyza I, Metcalfe S. Adaptivity and blow-up detection for nonlinear evolution problems. Journal on Scientific Computing. 2016; 38(6): A3833-56.
Moore PK. A posteriori error estimation with finite element semi-and fully discrete methods for nonlinear parabolic equations in one space dimension. SIAM Journal on numerical analysis.1994 Feb;31(1):149-69.
Douglas J, Dupont T. Galerkin methods for parabolic equations with nonlinear boundary Conditions. Numerische Mathematik. 1973 Jun 1; 20(3):213-37.
Cangiani A, Georgoulis EH, Jensen M. Discontinuous Galerkin Methods for Mass Transfer through Semipermeable Membranes. SIAM Journal on numerical analysis. 2013; 51(5):2911-34.
Cangiani A, Georgoulis EH, Jensen M. Discontinuous Galerkin methods for fast reactive mass transfer through semi-permeable membranes. Applied Numerical Mathematics. 2016 Jun 1; 104:3-14.
Metcalfe SA. Adaptive discontinuous Galerkin methods for nonlinear parabolic problems, PhD Thesis, University of Leicester 2015 Apr 10.
Sen Gupta J, Sinha RK. A posteriori error analysis of semilinear parabolic interface problems using elliptic Reconstruction. Applicable Analysis. 2018 Mar 12; 97(4):552-70.
Sabawi YA. Adaptive discontinuous Galerkin methods for interface problems, PhD Thesis, University of Leicester, Leicester, UK 2017.
Cangiani A, Georgoulis E, Sabawi Y. Adaptive discontinuous Galerkin methods for elliptic interface Problems. Mathematics of Computation. 2018; 87(314):2675-707.
Cangiani A, Georgoulis EH, Sabawi YA. Convergence of an adaptive discontinuous Galerkin method for elliptic interface problems. Journal of Computational and Applied Mathematics. 2020 Mar 15; 367:112397.
Sabawi YA. A Posteriori Error Analysis in Finite
Finite Element Approximation for Fully Discrete
Semilinear Parabolic Problems. InFinite
Element Methods and Their Applications 2020
Dec 10. IntechOpen.
Cangiani A, Georgoulis EH, Morozov AY, Sutton OJ. Revealing new dynamical patterns in a reaction– diffusion model with cyclic competition via a novel computational framework, Proceedings of the Royal Society A: Mathematical. Physical and Engineering Sciences. 2018 May 31; 474(2213):20170608.
Kyza I, Metcalfe S. Pointwise a posteriori error bounds for blow-up in the semilinear heat equation. SIAM Journal on Numerical Analysis. 2020;58(5):2609-31.
Sabawi YA. A Posteriori L_∞ (H^1 )- Error Bound in Finite Element Approximation of Semilinear Parabolic Problems. In2019 First International Conference of Computer and Applied Science (CAS) 2019 Dec 18; (pp. 102-106). IEEE.
Sabawi M. A Posteriori Error Analysis for Semidiscrete Semilinear Parabolic Problems. In2018 Al-Mansour International Conference on New Trends in Computing, Communication, and Information Technology (NTCCIT) 2018 Nov 14; (pp. 58-61). IEEE.
Sabawi M. A Posteriori Error Analysis for Fully Discrete Semilinear Parabolic Problems. In2018 2nd International Conference for Engineering, Technology and Sciences of Al-Kitab (ICETS) 2018 Dec 4 (pp. 22-27). IEEE.
Sabawi MA. Discontinuous Galerkin Timestepping for Nonlinear Parabolic Problems PhD Thesis, University of Leicester, Leicester, UK 2018.