New Features for PDE2D 9.2:

1. For symmetric 1D Galerkin problems, and symmetric 2D Galerkin problems with IDEG=-1 or -3, the time and memory required to find all eigenvalues of an eigenvalue problem are dramatically reduced. For one example 2D problem with N=3493 unknowns, the computer time decreased from 1491 seconds to 99 seconds. For an example 1D problem with N=3998 unknowns, the time decreased from 2582 seconds (or 1080 seconds with 4 processors) to 11 seconds, while memory decreased from 32.5 Mw to 0.3 Mw. The new algorithm is much faster because it can take advantage of the band structure of A in finding all eigenvalues of the generalized eigenvalue problem Az=λBz, but only when A is symmetric and B is diagonal (hence the requirement that IDEG=-1 or -3 for 2D Galerkin problems; if NEQN > 1, the RHO matrix must also be diagonal).

Notes:

1. Many important eigenvalue problems are symmetric, for example, (D(x,y)Ux)x + (D(x,y)Uy)y + a(x,y)U = λU is symmetric.
2. When only one eigenvalue is found, PDE2D has always taken advantage of the band or sparse structures of A and B, but when all eigenvalues are found, this is only possible for special cases such as above.
3. For more detail on the new algorithm, see page 30 of Appendix A .
   
   
2. For 3D problems, the MATLAB postprocessing program now draws the finite element grid in a 3D geometrically correct form, as well as the output grid.
3. For 1D and 2D Galerkin programs, when fixed and free boundary conditions are needed on the same arc, the fixed boundary condition Ui = FBi(X,...) can now be imposed by setting GBi = zero(Ui-FBi). It was already possible to mix fixed and free boundary conditions, this change just makes the documentation and usage clearer.
4. At the user's request, the values saved at each output point will be the solution as actually evaluated at a nearby collocation or integration point. For most problems this obviously will produce less accurate output or plots, but for certain (rare) problems (eg, the pressure in a fluid flow problem, with some formulations) a solution component may be much less noisy when plotted only at collocation or integration points.
5. For 1D Galerkin problems, the documentation and error checking for symmetric problems are much improved.