Fourier Spectral Moving Mesh Method for Willmore Equation of ...

area is not unchanged. Since we have no requirement for the boundaries, we cannot assumethe physical area is an unit square. Hence, it’s hard to propose the boundary conditions forthe Navier-Stokes equations. I have some ideas as follows:In order to force an unit square we need to force the displacement dx(x, y)| ∂Ω = 0 aswell as dy(x, y)| ∂Ω = 0. But we also require both dx and dy are periodic. And dx, dy arethe solutions of the MMPDEs. Hence, we cannot guarantee the existence of such a solution.And I need your suggestion on such issue.Another problem is about the semi-implicit scheme. Let’s review the equation(1 + ξ(λ ∗ κ 2 ) 3 δt)(ˆ¯φ − ˆφ) = δt{ ˙⃗x · ∇x φ + ξ∆ 3 φ − 1 ξ ∆2 W ′ (φ) − 1 ξ ∆(∆φW ′′ (φ)) +∫1ξ ∆(W ξ′(φ)W ′′ (φ)) + M[3 Ω 2 |∇φ|2 + 1 4ξ (φ2 − 1) 2 dx − β](−ξ∆ 2 φ + 1 ξ ∆W ′ (φ))} ∧I remembered in Hangzhou you told me that the splittingu t = a max ∆u implicit + (a(x) − a max )∆u explicitwill enhance the the stability. Hence I’m not surprised to see such a splitting scheme, whichutilized ∆ x φ = 1 ∇ J ξ · (JA∇ ξ φ) and it is some what like the splitting scheme above. And myquestion is that, why would such an semi-implicit scheme more stable? I use such a schemeover and over again, but I don’t know why? I failed to think it out. Is there some papers torefer to? Or would you mind explaining it to me?7 AppendixThe formulas in this paper have been cited many times,and here we give the detailed proof.a i = 1 J a j × a k , a i = Ja j × a k , a i · a j = δ i,jJ = a 1 · (a 2 × a 3 )Proof:First of all, by the definition of Jacobian Matrix it is obvious that J = a i · (a j × a k ) as longas i, j, k are in right hand order.Secondly, using the fundamental of inverse function, we have a i · a l = δil14