Uniform refinement

In order to compute functions which closely approximate limitingε-almost C-stationary points, we could use Algorithm 3 to solve the discrete penalised stationarity system (4.15) on a fine triangulation for increasingly large γ. This is computationally ex- pensive, as a solves on fine triangulations is required for each choice ofγ.

An alternative approach, which we state in Algorithm 4, is to use Algorithm 2 and increase γ at the same time as uniformly refining the triangulation. The motivation for this is the observation that h is sufficiently small by the time γ is large that (yγ_h, pγ_h) is always inside of the radius of convergence of the semismooth Newton method. The downside of the approach is that stationarity conditions such asyh≥0 may not be satisfied for a givenh.

Note that solving the discrete penalised stationarity system for a largeγ then uniformly refining the triangulation is not a viable strategy. In this case (yh, ph) is not a sufficiently good initial value for the Newton method without decreasingγ to a smaller value.

Algorithm 4 uses the function solvePen from Algorithm 2 to solve the discrete penalised stationarity system (4.15). The function refineAll outputs a uniform refinement of the current triangulation as well as the usual prolongations of

Algorithm 4Uniform refinement Input: Th, y0_h, p0_h and hmin, γmin>0

1: _data←(Ω, ν, f, a, b, I,{gω}ω∈I) 2: γ−←0 3: loop 4: γ ←γmin 5: while γ > chα do . c= 10, α= 2 6: (yγ_h, pγ_h)←solvePen(Th, γ, y_hγ−, pγ_h−,data) 7: γ−←γ, γ ←δγ·γ . δγ= 1.2 8: end while

9: if h > hmin then break

10: end if 11: (Th, yhγ, p γ h)←refineAll(Th, yhγ, p γ h) 12: end loop 13: Compute ξ_hγ, λγ_h according to (4.14), (4.15b) Output: y_hγ, pγ_h, ξ_hγ, λγ_h

the discrete functions so that that they are defined over the refined triangulation. In particular, it bisects all the triangular elements in such a way that the triangulation remains conforming.

We use the same value ofδγas in Algorithm 2 and use numerical experiments to calibrateα.

Calibrating α

Denote byuanε-almost limiting C-stationary point, byuh a solution of the discrete stationarity system (4.15), and byuγ_ha solution of the discrete penalised stationarity system (4.10).

We aim to find an exponent α such that for γ(h) =Chα we get the highest order convergence of ku−u_hγ(h)k_L2_(Ω) with respect toh. We do this by noting that

ku−uγ_h(h)k_L2_(Ω)≤ ku−u_hk_L2_(Ω)+ku_h−uγ(h)

h kL2_(Ω),

where the first term can be thought of as discretisation error and the second term can be thought of as penalisation error. So if we can findα1 and α2 such that

ku−uhkL2_(Ω)≤Chα1, ku_h−uγ_hk_L2_(Ω) ≤Cγ−α2,

withC independent ofhin both inequalities, then takingγ(h) =Ch−α1/α2 _{will give}

that

i.e. we should takeα=−α1/α2.

We will calibrateα1 andα2 using the following example. A discrete station-

arity point for this example is visualised in Figure 4.4.

Example 4.12. Let Ω = (0,1)2_{, A=−∆, ν = 0.003, f = 0, a=−100, b= 100,}

I ={(0.125,0.125),(0.125,0.5),(0.375,0.375).(0.5,0.125)},

andgω = 1 at (0.125,0.125) and gω= 0 otherwise.

First note thatkuh−uh0k_L2_(Ω)is a reasonable approximation toku_h−uk_L2_(Ω)

for h0 h. Calculating this quantity with h0 = 0.00552427 gives Figure 4.2(a). Based on this we speculate that

kuh−ukL2_(Ω)≤Chα1,

withα1 ≈1 and C independent ofh.

Next we calculate compute kuγ_h −uhkL2_(Ω), making sure that γ γ0 =

147789, the finalγ in Algorithm 3. For h= 0.0110485 this gives Figure 4.2(b), and we speculate that

kuγ,h−uhkL2_(Ω) ≤Cγ−α2,

with α2 ∈(1,2). Increasing γ is cheaper than decreasing h so we are conservative

and suppose α2 ≈ 1. As we have the function space convergence result Theorem

4.7, C should be roughly independent of γ. Combining these two experimentally observed convergence relationships we get that in order to stop the penalisation error dominating the discretisation error we should takeγ(h) = _hc for some constant

c.

We finish this section by testing our calibrated method on a simple example for which we know the exact solution, allowing the quantitieskuγ,h−u˜kL2_(Ω) and

Jˆ(˜u)−Jˆh(uh)

to be computed exactly. Taking c = 10 in this relationship gives

Figure 4.3.

We finish this section by testing the uniform refinement algorithm on a simple problem for which the solution is known exactly.

Example 4.13. Let Ω = (−1,1)2 and take

y(x) =    1 2(cos( 3 2π|x|) + 1) |x|< 2 3, 0 |x| ≥ 2₃.

(a) (b)

Figure 4.2: Convergence with respect to h and γ for Example 4.12.

Then y is the solution to variational inequality (4.1) with A = −∆ for f = −∆y

and u = 0. Now take ν = 1, b = a= −∞ and gω =y(ω) for all ω ∈ I. For any selection of points I the global minimiser of J is (y(0),0), as this is feasible and

J(y,0) = 0. We will take

I :={(0,0),(±0.5,±0.5),(0,±0.5),(±0.5,0)}.

We observe in Figure 4.3 the expectedO(h) convergence of kuγ,h−u˜kL2_(Ω)

and alsoO(h2) convergence of

ˆ J(˜u)−Jˆh(uh)

, where ˜uis a discrete stationary point

for smallh.