Numerical Analysis of Least-Squares Group Finite Element Method for Coupled Burgers' Problem
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Abstract
In this paper, a least squares group finite element method for solving coupled Burgers' problem in 2-D is presented. A fully discrete formulation of least squares finite element method is analyzed, the backward-Euler scheme for the time variable is considered, the discretization with respect to space variable is applied as biquadratic quadrangular elements with nine nodes for each element. The continuity, ellipticity, stability condition and error estimate of least squares group finite element method are proved. The theoretical results show that the error estimate of this method is . The numerical results are compared with the exact solution and other available literature when the convection-dominated case to illustrate the efficiency of the proposed method that are solved through implementation in MATLAB R2018a.
Received 1/8/2019
Accepted 15/8/2021
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Hussin C H C, Azmi A, Ismail A I M, Kilicman A. Approximate Analytical Solutions of Bright Optical Soliton for Nonlinear Schrödinger Equation of Power Law Nonlinearity, Baghdad Sci J., 2021; 18 (1), 836- 845.
Aswhad A A. Approximate Solution of Delay Differential Equations Using the Collocation Method Based on Bernstien Polynomials, Baghdad Sci J., 2011; 8(3), 820-825.
Barbara F S, Roberta V G, Paula C P, Romao E C. Interval study of convergence in the solution of 1D Burgers by least squares finite element method (LSFEM) + Newton linearization, Sci. Res. Essays, 2015; 10(16), 522-530.
Konzen P H A, Azevedo F S, Sauter E, Zingano P R A. Numerical simulations with the galerkin least squares finite element method for the Burgers’ equation on the real line, TEMA (São Carlos) , 2017; 18( 2), 287-304.
Ye X, Zhang S, A discontinuous least-squares finite-element method for second-order elliptic equations, Int. J. Comput. Math., 2018; ISSN: 1029-0265, 1-11.
Kalchev D Z, Manteuffel T A, Münzenmaier S. Mixed 〖(LL^*)〗^(-1) and LL^* least-squares finite element methods with application to linear hyperbolic problems. Numer. Linear Algebra Appl., 2018; special issue paper, 1-24.
Barrenechea G R, Knobloch P. Analysis of a group finite element formulation, Appl Numer Math., 2017; (18), 238–248.
Benosman M, Kramer B, Boufounos P T, Grover P. Learning-based Reduced Order Model Stabilization for Partial Differential Equations: Application to the Coupled Burgers’ Equation, American Cont. Conf., 2016; 1673-1678.
Noon N J. Weak Galerkin and Weak Group Finite Element Methods for 2-D Burgers' Problem, J Adv Res. Fluid Mech and Therm Sci ., 2020; 69(2), 42-59.
Fletcher C A J. Computational Techniques for Fluid Dynamics I, Springer-Verlag, 1991; 2th , ch.10, pp.360.
Cai Z, Manteuffel T A , Mccormick S F. First-order system least squares for velocity–vorticity–pressure form of the Stokes equations, with application to linear elasticity, Electron. Trans. Numer. Anal. , 1995; 3, 150–159.
Larson M G, Bengzon F. The finite element method: theory, implementation and applications, Texts in Computational Science and Engineering, Verlag Berlin Heidelberg, 2013; ISSN 1611-0994, 1th, ch.1, pp.6.
Ching L C, Yang S Y. Analysis of the [L2,L2,L2] least-squares finite element method for incompressible Oseen- type problems, Int. J. Numer. Anl. Mod., 2007; 4(3-4), (402-424)
Liew S C, Yeak S H. Multiscale Element-Free Galerkin Method with Penalty for 2D Burgers' Equation, J Teknol. (Sci & Eng), 2013; 62(3), 49-56.
Srivastava V K, Tamsir M, Bhardwaj U, Sanyasiraju Y. Crank-Nicolson scheme for numerical solutions of two-dimensional coupled Burgers' equations, JSER, 2011; 2 (5), 1-7.
Mittal R C, Tripathi A. Numerical solutions of two-dimensional Burgers' equations using modified Bi-cubic B-spline finite elements, Eng. Comput., 2015; 32 (5), 1275 -1306.
Kaennakham S, Chuathong N. A Proposed Adaptive Inverse Multiquadric Shape Parameter Applied with the Dual Reciprocity BEM to Nonlinear and Coupled PDE, J.Appl. Sci. , 2017; 17(10), 491-501.