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