Generalized Schwarzschild's method
  We describe a new finite element method (FEM) to construct continuous
equilibrium distribution functions of stellar systems. The method is a
generalization of Schwarzschild's orbit superposition method from the space of
discrete functions to continuous ones. In contrast to Schwarzschild's method,
FEM produces a continuous distribution function (DF) and satisfies the intra
element continuity and Jeans equations. The method employs two finite-element
meshes, one in configuration space and one in action space. The DF is
represented by its values at the nodes of the action-space mesh and by
interpolating functions inside the elements. The Galerkin projection of all
equations that involve the DF leads to a linear system of equations, which can
be solved for the nodal values of the DF using linear or quadratic programming,
or other optimization methods. We illustrate the superior performance of FEM by
constructing ergodic and anisotropic equilibrium DFs for spherical stellar
systems (Hernquist models). We also show that explicitly constraining the DF by
the Jeans equations leads to smoother and/or more accurate solutions with both
Schwarzschild's method and FEM.
