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.
Highlights
Molecular beam epitaxy (MBE) is a widely practiced technique for depositing atoms from a vapor phase onto a surface
We find that the rate of convergence in space is the fourth order in L2 norm and is the second order in H2 norm
6 Conclusion In this paper, the model is a class of fourth-order nonlinear parabolic differential equations with variable coefficient α(x, t)
Summary
Molecular beam epitaxy (MBE) is a widely practiced technique for depositing atoms from a vapor phase onto a surface. In the limit of weak desorption, Pierre-Louis et al derived a 1D case of equation (3) for vicinal surface growing in the step flow mode [11] This limit turned out to be singular, and nonlinearities of arbitrary order need to be taken into account. For 1D case of equation (3), Zhao et al proved that the Hermite finite element method has the convergence rate of O( t + h3) (see [12]). The finite element method (FEM) [13,14,15,16,17,18], as a type of an important numerical tool for solving differential equations, has a long history. We consider the cubic B-spline FEM for a fourth-order nonlinear parabolic equation with variable coefficient. There exists a unique global solution u(x, t) for problem (8) such that u ∈ L∞ 0, T; H02(I) ∩ L2 0, T; H4(I) , ut ∈ L2 0, T; L2(I)
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