### ERROR ESTIMATES FOR TWO-SCALE COMPOSITE FINITE ELEMENT APPROXIMATIONS OF NONLINEAR PARABOLIC EQUATIONS

Tamal Pramanick

1. Department of Mathematics, National Institute of Technology Calicut, Kozhikode-673601, India
• Received:2019-05-05 Revised:2019-10-09 Published:2021-08-06
• Contact: Tamal Pramanick,Email:t.pramanick002@gmail.com
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The author would like to express his sincere thanks to the anonymous referees for their helpful comments and suggestions, which greatly improved the quality of this paper. He is grateful to Professor Rajen Kumar Sinha, Indian Institute of Technology Guwahati, India for discussing the results presented in this article and inspiring for this research. The author gratefully acknowledges valuable support provided by the Department of Mathematics, NIT Calicut and the DST, Government of India, for providing support to carry out this work under the scheme ‘FIST’ (No. SR/FST/MS-I/2019/40).

Tamal Pramanick. ERROR ESTIMATES FOR TWO-SCALE COMPOSITE FINITE ELEMENT APPROXIMATIONS OF NONLINEAR PARABOLIC EQUATIONS[J]. Journal of Computational Mathematics, 2021, 39(4): 493-517.

We study spatially semidiscrete and fully discrete two-scale composite finite element method for approximations of the nonlinear parabolic equations with homogeneous Dirichlet boundary conditions in a convex polygonal domain in the plane. This new class of finite elements, which is called composite finite elements, was first introduced by Hackbusch and Sauter [Numer. Math., 75 (1997), pp. 447-472] for the approximation of partial differential equations on domains with complicated geometry. The aim of this paper is to introduce an efficient numerical method which gives a lower dimensional approach for solving partial differential equations by domain discretization method. The composite finite element method introduces two-scale grid for discretization of the domain, the coarse-scale and the fine-scale grid with the degrees of freedom lies on the coarse-scale grid only. While the fine-scale grid is used to resolve the Dirichlet boundary condition, the dimension of the finite element space depends only on the coarse-scale grid. As a consequence, the resulting linear system will have a fewer number of unknowns. A continuous, piecewise linear composite finite element space is employed for the space discretization whereas the time discretization is based on both the backward Euler and the Crank-Nicolson methods. We have derived the error estimates in the L(L2)-norm for both semidiscrete and fully discrete schemes. Moreover, numerical simulations show that the proposed method is an efficient method to provide a good approximate solution.

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