tailieunhanh - Linearized four-step implicit scheme for nonlinear parabolic interface problems

We present the solution of a second-order nonlinear parabolic interface problem on a quasiuniform triangular finite element with a linearized four-step implicit scheme used for the time discretization. The convergence of the scheme in L2-norm is established under certain regularity assumptions using interpolation and elliptic projection operators. | Turk J Math (2018) 42: 3034 – 3049 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article Linearized four-step implicit scheme for nonlinear parabolic interface problems 1 Matthew Olayiwola ADEWOLE1,∗,, Victor Folarin PAYNE2 Department of Computer Science and Mathematics, Mountain Top University, Prayer City, Ogun State, Nigeria 2 Department of Mathematics, University of Ibadan, Ibadan, Oyo State, Nigeria Received: • Accepted/Published Online: • Final Version: Abstract: We present the solution of a second-order nonlinear parabolic interface problem on a quasiuniform triangular finite element with a linearized four-step implicit scheme used for the time discretization. The convergence of the scheme in L2 -norm is established under certain regularity assumptions using interpolation and elliptic projection operators. A numerical experiment is presented to support the theoretical result. It is assumed that the interface cannot be fitted exactly. Key words: Four-step implicit, interface, almost-optimal, nonlinear parabolic equation 1. Introduction Parabolic interface problems are frequently encountered in scientific computing and industrial applications. A typical example is provided in the modeling of heat diffusion, which involves two or more materials with different properties [8]. The most well-known linear parabolic partial differential equation (PDE) is the heat equation. However, the linear heat equation has some limitations that could be addressed by nonlinear generalizations [5]. It is therefore necessary to investigate the solution of nonlinear PDEs on bounded domains. The problem becomes an interface problem when more than one material medium with different properties is involved. Many contributions have been made towards the development of the finite element method (FEM) for linear parabolic interface problems, ., [2, 4, 13–16, 21]. Semilinear .