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Abstract
The fourth-order nonlinear Sivashinsky equation is often used to simulate a planar solid-liquid interface for a binary alloy. In this paper, we study the high accuracy analysis of the nonconforming mixed finite element method (MFEM for short) for this equation. Firstly, by use of the special property of the nonconforming EQ1rot element (see Lemma 1), the superclose estimates of order Oh2+Δt in the broken H1-norm for the original variable u and intermediate variable p are deduced for the back-Euler (B-E for short) fully-discrete scheme. Secondly, the global superconvergence results of order Oh2+Δt for the two variables are derived through interpolation postprocessing technique. Finally, a numerical example is provided to illustrate validity and efficiency of our theoretical analysis and method.
Highlights
Under certain conditions, the dilute binary alloy will solidify, at which point the solid-liquid interface is unstable and has a cellular structure
As it is known to all that in regard to the fourth-order problem, the conforming Galerkin finite element (FE for short) approximation space belongs to H2(Ω), and FE solution in turn shall be C1-continuous. is leads to the higher degree of piecewise polynomials, and the related computation is complicated and difficult. e MFEM is an optimal choice to overcome the above deficiencies, which transforms a fourth-order problem into 2 coupled second-order problems
Mathematical Problems in Engineering by introducing an intermediate variable; the low-order elements can be used to solve. e nonconforming MFEM brings down the smoothness requirement on FE solution compared to the conforming case
Summary
The dilute binary alloy will solidify, at which point the solid-liquid interface is unstable and has a cellular structure. As it is known to all that in regard to the fourth-order problem, the conforming Galerkin finite element (FE for short) approximation space belongs to H2(Ω), and FE solution in turn shall be C1-continuous. E main purpose of this article is to develop a nonconforming MFE scheme for problem (1), and the superclose and superconvergence results of the original variable u and auxiliary variable p in the broken H1-norm are obtained for the B-E fully-discrete scheme.
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