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Do you know any methods/solvers for PDE or eigenvalue problems like

  • $\begin{cases} \Delta u= 0\ (\text{ or }\lambda u) & \text{ in }\Omega \\ u =0 & \text{ on }\partial \Omega \end{cases}$ (Dirichlet)
  • $\begin{cases} \Delta u= 0\ (\text{ or }\lambda u) & \text{ in }\Omega \\ \frac{\partial u}{\partial n} =0 & \text{ on }\partial \Omega \end{cases}$ (Neumann)
  • $\begin{cases} \Delta u= 0\ (\text{ or }\lambda u) & \text{ in }\Omega \\ \frac{\partial u}{\partial n}+\beta u =0 & \text{ on }\partial \Omega \end{cases}$ (Robin)

in 2D and 3D that do not involve the Finite Element Method?

In 2D there is at least an option (for Dirichlet and Neumann): mpspack which uses a basis for the space of harmonic functions and works only with coefficients in this basis. In 3D I do not know anything that doesn't use Finite Element Method.

I am interested in a different method than FEM because in shape optimization domains change at every iteration and so does the mesh, the discretization, etc.

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It seems to me that with some straightforward mesh moving/updating techniques, the Finite-element Method is the most likely to be successful on optimization problems where the shape evolves somewhat smoothly. If the topology of the mesh changes (folds on itself, adds/delete holes, etc.) then the meshing problem is harder, but there are lots of folks doing shape optimization with FEM. –  Bill Barth Jan 11 '13 at 23:50
    
A myriad of local and global volumetric (integral and differential) methods are available, as well as boundary integral methods. Among the local, volumetric, differential methods, some are based on meshes (FEM is a subset of this class) and some are not. You reference mpspack which is a least squares finite element method (using basis functions that satisfy the PDE inside each element, but are not continuous between elements) primarily targeted at scattering problems (Helmholtz). Can you elaborate on the domains you are interested in? –  Jed Brown Jan 12 '13 at 1:00
    
My current work is mostly related to eigenvalue optimization problems with respect to different constraints. –  Beni Bogosel Jan 12 '13 at 9:35
    
@BeniBogosel You have to be more specific than that. Are you doing topology optimization? Are the boundaries smooth? Reentrant corners? –  Jed Brown Jan 12 '13 at 19:32
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@BeniBogosel I'm not aware of any that are similar to mpspack, but you could use pure boundary integral methods. BEM++ has a Python interface. bempp.org –  Jed Brown Jan 18 '13 at 22:37

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