Why should the spatial derivative order of the ODE *not* exceed two?

Following this question I came across this strange behaviour.

Let me define a 1 D interval implicitely

a = 1; b = 2;
Ω = ImplicitRegion[a <= r <= b, {r}]

and let me try to solve PDEs on this interval.

This works as expected (second order ODE)

w1 = NDSolveValue[{w''[r] == 1/2,
DirichletCondition[w[r] == 0, r == a],
DirichletCondition[w[r] == 0, r == b]}, w, {r, a, b}]

Plot[w1[x], x ∈ Ω] and so does this

w2 = NDSolveValue[{w''[r] == 1/2,
DirichletCondition[w[r] == 0, r == a],
DirichletCondition[w[r] == 0, r == b]}, w, r ∈ Ω]

This works again as expected (this time 4th order ODE)

w1 = NDSolveValue[{w''''[r] == 1/2,
DirichletCondition[w[r] == 0, r == a],
DirichletCondition[w'[r] == 0, r == a],
DirichletCondition[w[r] == 0, r == b],
DirichletCondition[w'[r] == 0, r == b]}, w, {r, a, b}]

Plot[w1[x], x ∈ Ω] Question

But why does this fail?

w2 = NDSolveValue[{w''''[r] == 1/2,
DirichletCondition[w[r] == 0, r == a],
DirichletCondition[w'[r] == 0, r == a],
DirichletCondition[w[r] == 0, r == b],
DirichletCondition[w'[r] == 0, r == b]}, w, r ∈ Ω]

with the error message The spatial derivative order of the PDE may not exceed two. >>

The only difference being using r ∈ Ω as a domain instead of {r,a,b}.

• I think it may be a limitation of the FEM code. – Michael E2 Aug 8 '15 at 17:18
• @MichaelE2 But why does it switch to the FEM code if the default code works? – chris Aug 8 '15 at 17:20
• I was just about to add: Because in the non-working code, you use regions. -- That is to say, the use of regions causes NDSolve to go to FEM, without looking at whether the region is amenable to other methods. – Michael E2 Aug 8 '15 at 17:20
• @MichaelE2 it is a pity though is it not? – chris Aug 8 '15 at 17:22
• It's probably irrelevant, but conceptually spatial regions have no innate direction but time does. That fact that Ω could be viewed as having a direction and therefore integrated over as if it were time is a coincidence of it being 1D and the necessity of describing it in terms of a coordinate system. Still, you'd think it could say "FEM failed; integrating as an ODE with temporal variable r." – Michael E2 Aug 8 '15 at 18:35