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numerical integration - Can you use "MaxErrorIncreases" when numerically integrating over a region?



How do you pass the option "MaxErrorIncreases" to NIntegrate[] when integrating over a region?


Taking a simple ;) example from Problems with NIntegrate, errors NIntegrate::slwcon and NIntegrate::inumri:


Ω = ImplicitRegion[
   0 <= x <= 34 && -18 <= y <= 0 && ! ((12 < x < 15 && -12 < y) ||
    (15 < x < 20 && -3 < y) || (20 < x < 23 && -7 < y) ||
    (23 < x <= 34 && -3 < y)), {x, y}];
fi = NDSolve[{
   Derivative[0, 2][φ][x, y] + Derivative[2, 0][φ][x, y] == NeumannValue[0, 
     x == 0 && -18 < y < 0 || 0 < x < 34 && y == -18 || 
      x == 34 && -18 < y < -3 || x == 23 && -7 < y < -3 || 

      20 < x < 23 && y == -7 || x == 20 && -7 < y < -3 || 
      15 < x < 20 && y == -3 || x == 15 && -12 < y < -3 || 
      12 < x < 15 && y == -12 || x == 12 && -12 < y < 0], 
   DirichletCondition[φ[x, y] == 1.5, 0 <= x <= 12 && y == 0], 
   DirichletCondition[φ[x, y] == 0.4, 23 <= x <= 34 && y == -3]},
   φ, {x, y} ∈ Ω,
  "ExtrapolationHandler" -> {0 &, "WarningMessage" -> False}];

Then on this integral we get a warning that suggests raising "MaxErrorIncreases":


ClearAll[ff];

dxφ = Derivative[1, 0][φ] /. First[fi];
ff[x_?NumericQ, y_?NumericQ] := Quiet@Check[dxφ[x, y], 0.];
NIntegrate[ff[x, y], {x, y} ∈ Ω]


NIntegrate::eincr: The global error of the strategy GlobalAdaptive has increased more than 2000 times....Increasing the value of the GlobalAdaptive option MaxErrorIncreases might lead to a convergent numerical integration....



However, using the usual Method setting results in an error:


NIntegrate[ff[x, y], {x, y} ∈ Ω, Method -> {"GlobalAdaptive", "MaxErrorIncreases" -> 5000}]



NIntegrate::regm: Method GlobalAdaptive is not applicable for a region domain. Continuing with method Automatic.



Is it possible to use "MaxErrorIncreases" with NIntegrate[] over a region?



Answer



I found the strategy, "SymbolicDomainDecomposition":


NIntegrate[ff[x, y], {x, y} ∈ Ω, 
 Method -> {"SymbolicDomainDecomposition", 
   Method -> {"GlobalAdaptive", "MaxErrorIncreases" -> 5000}}]



NIntegrate::eincr: The global error of the strategy GlobalAdaptive has increased more than 5000 times....



(*  -6.0019  *)

It turns out not to be good way to integrate this function. Even with a setting of "MaxErrorIncreases" -> 100000, the result still has a large error. There are better methods for the example integral in the OP, which may be found in the linked question.


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