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I have been trying to run NDEigensystem on a differential operator with four variables $\alpha$, $\beta$, $\gamma$, and $\delta$. I've found that if I run the exact same code with one less variable, it works perfectly. But when I try with 4 dimensions, I get a cryptic error message:

NDEigensystem::fem3 --Message Text Not Found--

Is this because NDEigensystem simply doesn't work in dimensions greater than three? If so, is there any known way around this?

Here's my code:

HPenta = -Laplacian[ψ[α, β, γ, δ], {α, β, γ, δ}];
regionPenta = Simplex[{1/4 {Sqrt[5], Sqrt[5], Sqrt[5], -1}, 
    1/4 {Sqrt[5], -Sqrt[5], -Sqrt[5], -1}, 
    1/4 {-Sqrt[5], Sqrt[5], -Sqrt[5], -1}, 
    1/4 {-Sqrt[5], -Sqrt[5], Sqrt[5], -1}, {0, 0, 0, 1}}];

wfsPenta = 
 NDEigensystem[{HPenta, 
   DirichletCondition[ψ[α, β, γ, δ] ==0, True]}, 
   ψ[α, β, γ, δ], {α, β, γ, δ} ∈ regionPenta, 12]
user21
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NOABM
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  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Mar 20 '24 at 14:48
  • Welcome to Mathematica StackExchange! Please copy and paste the Mathematica code you are using, so that others can reproduce your issue and give you concrete feedback on how to correct the code :) – Domen Mar 20 '24 at 15:44
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    Please look at the comments on the following question NDEigensystem works for 3-D Schrodinger Equation but not 4-D. It seems that the current implementation of FEM supports only up to 3 spatial dimensions. However, I agree that the missing error message should be reported to the WRI. – Domen Mar 20 '24 at 16:24
  • Also {\[ScriptCapitalL], \[ScriptCapitalB]} = {-Laplacian[ u[x, y, z, t], {x, y, z, t}], DirichletCondition[u[x, y, z, t] == 0, True]};{vals, funs} = NDEigensystem[{\[ScriptCapitalL], \[ScriptCapitalB]}, u[x, y, z, t], {x, 0, \[Pi]}, {y, 0, \[Pi]}, {z, 0, \[Pi]}, {t, 0, \[Pi]}, 2] performs "Set::shape: Lists {vals,funs} and NDEigensystem[{-(u^(0,0,0,2))[x,y,z,t]-(u^(0,0,2,0))[x,y,z,t]-(u^(0,2,0,0))[x,y,z,t]-(u^(2,0,0,0))[x,y,z,t],DirichletCondition[u[x,y,z,t]==0,True]},<<5>>,2] are not the same shape.". – user64494 Mar 20 '24 at 16:34
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    @user64494, your code does not add any additional value; the error message is obvious, since NDEigensystem remains unevaluated, yet you want it to assign to two different symbols. – Domen Mar 20 '24 at 16:45
  • Domen (@ does not work}: Adds: Cuboid differs from Simplex. – user64494 Mar 20 '24 at 16:56

2 Answers2

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No, the finite element method in the Wolfram language is implemented up to 3D. Since NDEigensystem and friends are based ilin the FEM, the answer is no.

user21
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-2

The growth of memory by power of space dimension $n$ and time by evaluation of $n*m$-dimensional kernels over $n$-dimensional subsets of length $m$ in each dimension, you can roughly check, that a problem in 1d with 1000 points is growong to 10^6 points in 2d and 10^9 points in 3d.

If time is involved, you are working on grids of 10^12 points, typically with a kernel of 3^4 =81 elements for second order PDE's. From these numbers you can evaluate the price of the GPU to be used to produce 1 result per day by the number of parallelly working kernels on a linear parallelizable problem.

In these days, the IT-advanced communities are struggling to build enough wind power stations per day in order to feed the cloud, the bitcoin industry and the video streaming transfers, for nothing more than an increase the the overall noise level, aka temperature of sea and atmosphere.

Roland F
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