Strict tolerance checks for CompositeGridFunctionEstimator

[anaegel]

The current implementation (as of 3c01ad1) of CompositeGridFunctionEstimator accepts a step, if
(1)

$$ \sum_i |\delta|_i^2 \le TOL^2 \sum_i |u|_i^2 $$

In some cases, a stricter tolerance is desirable. In particular, we may want to enforce the tolerance in in a component-wise fashion:
(2)

$$\max_i \frac { |\delta|_i } { |u|_i } \le TOL$$

I suggest to a add function 'use_strict_relative_norms' to enforce this.

=> Note that (2) induces (1)!

[anaegel]

Commits 4925273 and 16e65f2 address this. Please feel free to comment @LogashenkoDL, @lus0a, @junxiW, @Nordegraf and everybody ;-)!

[anaegel]

Here's a small example:

local limexErrorEst = CompositeGridFunctionEstimator()
local weightedMetricSpace = CompositeSpace()

local scalePSquared = 1.0
local scaleCSquared = 1.0
local spaceP = VelEnergyComponentSpace("p", 2, ConstUserMatrix(scalePSquared))
local spaceC = L2ComponentSpace("c", 2, scaleCSquared)

weightedMetricSpace:add(spaceC)
weightedMetricSpace:add(spaceP)

limexErrorEst:add(weightedMetricSpace)
limexErrorEst:use_strict_relative_norms(true)

[Nordegraf]

This improved the time step sizes for the lens testcase greatly! I'm running more simulations with more refinements, but for up to 7 refinements this is a clear improvement.

[anaegel]

This improved the time step sizes for the lens testcase greatly! I'm running more simulations with more refinements, but for up to 7 refinements this is a clear improvement.

@Nordegraf: Thx for the feedback! Note that, if required, you can now also try to include the H1 semi-norm for the concentration:

local spaceC2 = H1SemiComponentSpace("c", 2)
weightedMetricSpace:add(spaceC2)

No additional scaling is required!

[anaegel]

One drawback of

$$\max_i \frac { | \delta |_i } { |u|_i } \le TOL$$

is, that it is ill-posed for $| \delta|_i \approx |u|_i \approx 0$. Alternatively, one can use

$$\max_i \frac { | \delta |_i^2 } { | u |_i^2 + \epsilon \frac{1}{1+ | u |_i^2 + | \delta |_i^2 }} \le TOL^2$$

which yields

$$ \sum_{i=1}^n | \delta |_i^2 \le TOL^2 ( \sum_i |u |_i^2 + \epsilon n ) $$

This is realized in 2a5c62a.

[lus0a]

Thanks for these great improvements! It works well for my cases. Not only getting rid of the scaling coefficients but also guaranteeing accuracy when the component norm is near zero.