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In FEMAddOns there is a function DecompositionNDSolveValue to solve stationary PDEs on a cluster. Is there a similar way to solve the eigenvalue problem for a system of ODEs in parallel kernels, something like DecompositionNDEigenvalues?

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No, unfortunately there is not. The domain decomposition solver DecompositionNDSolveValue you mention from the FEMAddOns works by decomposing the domain, solving the PDE on the subdomain and make use of this subdomain solution as a boundary condition for the solution on another subdomain in clever way. This approach does not work for NDEigensystem or Eigensolvers for the simple fact that they do not make use of boundary conditions at all. It's the same reason why NDEigensystem does not work for homogeneous Dirichelt or NeumannValue boundary conditions (NDEigensystem does work for 'homogeneous' Robin BCs though).

Also note, that the DomainDecompositionSover is most likely not faster then local solvers - the main point of the DDS is to solve large scale PDEs where memory is the main constraint.

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