A NOTE ON CORRECTORS FOR THE SEMI-LINEAR ELLIPTIC EQUATION


In the previous note, we proved new results on the high-order correctors for the semi-linear elliptic systems associated with the nonlinear interior boundary conditions. Such interesting results give us a deep insight into the convergence rate of the formal asymptotic expansions at a general level. In this note, we merely take into consideration the semi-linear equation with zero boundary conditions. Despite its simple form, the important thing is to show the case that the so-called auxiliary problems are nonlinear. At this point we may see, applying formal homogenization procedures is definitely impossible. Henceforward, a convergent scheme for the nonlinear auxiliary problems is needed.

The semi-linear elliptic equation being considered has the form

\nabla\cdot(-d^{\varepsilon}\nabla u^{\varepsilon})=R(u^{\varepsilon}),

where the reaction term is supposed to be globally L-Lipschitzian (L is independent of the homogenization parameter \varepsilon) and the diffusion term d^{\varepsilon} is the same as the previous note.

Using the formal asymptotic expansion at Mth-order, we introduce that

u^{\varepsilon}=\displaystyle{\sum_{m=0}^{M}\varepsilon^{m}u_{m}(x,y) + \mathcal{O}(\varepsilon^{M+1})}.

We remind that the functions u_{m} is formally the solutions of the auxiliary problems. In the linear case, it is possible to solve such problems but the nonlinear case may exist if we have

\displaystyle{R(\sum_{m=0}^{M}\varepsilon^{m}u_{m})=\sum_{m=0}^{M}\varepsilon^{m-r}R(u_{m})+\mathcal{O}(\varepsilon^{M-r+1})},

for r\in\mathbb{Z} and r\le 2.

In particular, taking r=2 the impediment stems from the following nonlinear (more precisely, semi-linear) auxiliary problems, and that should be passed without difficulties since solving nonlinear elliptic problems did turn to be easy-to-find many decades ago. Among many methods, we are strongly interested in the so-called linearization due to its fame as well as its usage. On the whole, this method and its modified versions are applied to various kind of PDEs. For example, it is devoted to dealing with the Kirchhoff-Carrier type wave equation in [Long2002] or very recently approximates the solution of semi-linear ultra-parabolic equations in [Khọa2015] under Fourier-mode and zigzag-FDM. Certainly, the procedures start from the Picard-based iteration and it is linearly convergent in the sense of discrete consideration. Employing this approach, we totally solve the nonlinear auxiliary problems, but must have a mild restriction whilst the convergence rate is undoubtedly the same with the standard results.

In addition from this note that, the corrector herein still inherits from the recent result in [Khoa2016] where the order \mathcal{O}(\varepsilon^{M-1}) is obtained.

All of the above-mentioned results could be found in the publication attached here.

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