Generic elliptic solver

Notes

Requirements:

• Steve: FunWave, need parallelism and mesh refinement;
• Ian: 6-variable, linear elliptic equation, may need mesh refinement and definitely parallelism;
• Eloisa: generic solver, easy to use and to experiment with more important than efficiency, no restriction on topology, mesh refinement would be good but doesn't need it for everything;
• Do these requirements limit the type of solver? Can we use multigrid? Spectral?
• Boundary conditions play now a more important role than for hyperbolic problems. Are the current mechanisms for specifying them enough? Do we need, say, inner boundaries?

Existing tools:

• Scott's multigrid elliptic solver, second order with extension to fourth order coming soon; integrate with Cactus via own data conversions, but currently making it talk to Carpet. Available immediately via SVN;
• Eloisa has experimented with OpenFOAM: nice and flexible, not great for accuracy, need to import data to Cactus afterwards (not complicated, but unfeasible to do at each timestep).
• Current CactusElliptic: EllBase gives interface to register elliptic solvers, currently not much implemented (SOR), equation type is a little restrictive (linear), compatibility with AMR unknown;
• CarpetPETSc (can PETSc work with AMR)?
• Existing implementations:
  • TwoPunctures;
  • NoExcision (FD conjugate-gradient, system of decoupled equations, no AMR);
  • Kranc-generated Laplace solver via relaxation;
  • BAM_Elliptic (https://svn.aei.mpg.de/numrel/AEIThorns/BAM_Elliptic/trunk/; usability? license?);
  • TATElliptic (interface only)?
  • Aaryn's solver (2D)?
• There's currently nothing that allows for AMR!
• Chombo multigrid AMR?

Possible directions:

• EllBase road, make type of equation more generic:
  • Add more terms and coefficients;
  • Ian: specify the equation via a callback for the residual, a registration method for the variables to solve for and possibly the derivatives/jacobians, much like in MoL (these could be generated via Kranc); this would also really only work with relaxation methods, not for direct inversion methods. Many solvers, all working along these guidelines? EllBase would provide the interface;