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.
Highlights
Nonlinear partial differential equations arise in many fields of science, in physics and engineering [1,2]
It can be seen that the LSGrFEM performed better results and agree with exact solutions than that obtained by literature 16, for which the exact solution is given as, u(x, y, t)
Theoretical analysis show that the error estimate of LSGrFEM is O(hr)
Summary
Nonlinear partial differential equations arise in many fields of science, in physics and engineering [1,2]. Ye and Zhang 5 applied the discontinuous least-squares (DLS) finite element method to second-order elliptic equations. Theoretical error estimates were presented and numerical solutions were given to demonstrate the accuracy approximate of this method. Manteuffel and Münzenmaier 6 studied Mixed (LL∗)−1 and (LL∗) least-squares finite element methods with application to linear hyperbolic problems. They were founded upon and extended the LL∗ approach that is rather general and applicable beyond the setting of elliptic problems. The least squares group finite element method for 2-D coupled Burgers’ problem with a fully-discrete approximation for the time variable is presented. Let C7 and C8 are positive constants independent of h the following error estimates are hold : max ‖un. For any φh , ψh ∈ Vh choosing them such that the approximation properties 13 – 14 are satisfied, when φ and ψ are replaced by unand vn ,
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