Adaptivity with moving grids

50 C. J. Budd, W. Huang and R. D. Russell
R ≪ 1andR(a) =R(b) = 0. To leading order, R satisfies the equation
ɛ(Ṙ − γR ξξ) =(MR ξ ) ξ +(M x x ξ R) ξ =(MR ξ + M ξ R) ξ =(MR) ξξ .
Therefore
ɛṘ =(1− γ∂2 ξ )−1 (MR) ξξ ≡ G(MR) ξξ . (3.14)
Here G is a positive compact operator and, as M>0, ER ≡ (MR ξξ )isa
uniformly elliptic operator with a negative real spectrum. It follows that R
must decay to zero. Hence the equidistributed solution is locally stable.
(ii) To prove this result, consider a slowly varying monitor function M(x, t)
and an exact solution ˆx of the equidistribution equation (M ˆx ξ ) ξ =0. If
x =ˆx + ɛR is a solution of (3.9) then, extending the calculation in (3.14),
we see that R satisfies the equation
ɛ(ẋ − γẋ ξξ )+O(ɛ 2 )=ɛ(MR) ξξ + O(ɛ 2 ).
Hence, we have
(MR) ξξ = ˙ˆx − γ ˙ˆx ξξ + O(ɛ).
But as (M ˆx ξ ) ξ =0,wehave
(M ˙ˆx ξ ) ξ = −(M tˆx ξ ) ξ .
As the operator Eφ ≡ (Mφ) ξξ is uniformly elliptic, it follows that provided
Ṁ is of order one, then ˙ˆx and hence R and its derivatives are also of order
one. Consequently, the solution x of (3.9) stays ɛ-close to the solution of
the equidistribution equation.
(iii) To show this we need to show that x ξ cannot vanish. This result is a
consequence of the maximum principle. In the case when γ =0,wehave,
on differentiating (3.9), that
ẋ ξ = M ξξ x ξ +2M ξ x ξξ + Mx ξξξ .
Suppose that x ξ is initially positive everywhere, and as x evolves it vanishes
for a first time at (without loss of generality) the point ξ = 0. Then locally
close to this point we have x ξ = aξ 2 + O(ξ 3 ) for some a>0. Hence
ẋ ξ = aξ 2 M ξξ +2aξM ξ + aM + O(ξ).
Thus ẋ ξ > 0 at this point and time. Hence x ξ must remain positive. The
more general result follows from the positivity of the compact operator G.
3.1.1. Coupling a one-dimensional MMPDE method to a PDE
In one dimension, MMPDE methods can be very effectively coupled to an
underlying PDE system by using a variety of different methods, including
finite difference, finite element, collocation and spectral methods. The