A B-spline finite element method for nonlinear differential equations describing crystal surface growth with variable coefficient

Abstract

In this paper, an efficient finite element scheme is presented for a class of fourth-order nonlinear parabolic problems with variable coefficient. To deal with second-order term in weak formulation, we choose the cubic B-spline function as a trial function. Rigorous error estimates are derived for both semi-discrete and fully-discrete schemes. We provide a numerical example to confirm our theoretical results.

A cubic B-spline finite element method for a class of fourth order nonlinear differential equation with variable coefficient

“…be the basis functions of FEM. In order to deal with the boundary conditions, we modify the boundary B-spline basis functions according to [14,15]. The approximate solution can be written as follows:…”

“…The finite element method (FEM) is effective in solving partial differential equations [9][10][11]. Some papers, which have already been published, study the Cahn-Hilliard equations using various different forms of FEM [14][15][16][17][18]. In [14,15], Qin et al considered two different fourth order nonlinear parabolic problems with variable coefficient employing B-spline FEM respectively, and the boundness and the error estimates of the approximate solutions were proved.…”

“…Some papers, which have already been published, study the Cahn-Hilliard equations using various different forms of FEM [14][15][16][17][18]. In [14,15], Qin et al considered two different fourth order nonlinear parabolic problems with variable coefficient employing B-spline FEM respectively, and the boundness and the error estimates of the approximate solutions were proved. The nonlinear term and the fully discrete scheme in [14] were different from that of [15].…”

“…In [14,15], Qin et al considered two different fourth order nonlinear parabolic problems with variable coefficient employing B-spline FEM respectively, and the boundness and the error estimates of the approximate solutions were proved. The nonlinear term and the fully discrete scheme in [14] were different from that of [15]. Bao et al conducted numerical experiments to study the effect of a precursor fluid layer on the motion of two phase system in a channel [16].…”

“…For example, cubic B-splines are in C 2 (−∞, +∞). However, the B-spline basis functions need to be modified to deal with boundary conditions [14,15].…”