Adaptivity with moving grids

Adaptivity with moving grids 69
augmented Lagrangian approach and a least-squares formulation respectively.
Feng and Neilan (2009) consider the nonlinear second-order equation
as the limiting equation of a singularly perturbed fourth-order quasilinear
equation. Chartrand, Vixie, Wohlberg and Bollt (2007) show that
the Monge–Ampère equation can be reformulated as an unconstrained optimization
problem, which can be solved by a gradient descent method.
A special property of the mappings generated by the Monge–Ampère equation
is that they are irrotational, i.e., the curl is zero. Haker and Tannenbaum
(2003) propose a gradient descent method that uses a Poisson solve
in each step to ‘remove the curl’. The nonlinear multigrid method developed
in Fulton (1989) for the semigeostrophic equation should be easily
adapted to our problem. A Newton–Krylov-multigrid method is proposed
in Delzanno et al. (2008). The above methods all try to solve the fully
nonlinear Monge–Ampère equation directly. It was a remarkable achievement
of Kantorovich to show that the problem can be relaxed to a linear
one by considering not a transport map ξ → x = F(ξ), but a transport
plan G(ξ, x) indicating the amount of material to be transported from
ξ to x (Rachev and Rüschendorf 1998, Evans 1999). Robust methods
exist for solving the corresponding linear programming problem, but to
the best of our knowledge these methods typically require O(N 2 ) operations
(Kaijser 1998, Balinski 1986), which is unacceptable except for small
problems.
The parabolic Monge–Ampère (PMA) method. An alternative approach motivated
by the discussion of the MMPDEs given earlier, is to introduce a
parabolic regularization to (3.41) so that the gradient of solutions of this
evolve toward the gradient of the solutions of (3.41) over a (relatively) short
time scale. This method also couples naturally to the solution of a timedependent
PDE. Accordingly we consider using relaxation to generate an
approximate solution of (3.41), which evolves together with the solution
of the underlying PDE. Accordingly we consider a time-evolving function
Q(ξ,t) with associated mesh X(ξ,t)=∇ ξ Q(ξ,t), with the property that
this mesh should be close to that determined by the solution of the Monge–
Ampère equation. To do this we consider a relaxed form of (3.41) taking
the form of a parabolic Monge–Ampère equation (PMA) of the form
ɛ(I − γ∆ ξ )Q t = ( H(Q)M(∇ ξ Q) ) 1/n . (3.46)
To find a moving mesh, we start with an initially uniform mesh for which
Q(ξ, 0) = 1 2 |ξ|2 .
The function Q then evolves according to (3.46). In (3.46) the scaling power
1/n is necessary for global existence of the solution. This is because if Q
is scaled by a factor L(t) then the Hessian term H(Q) scales as L(t) n .