  A moving mesh finite element method is studied for the numerical solution of
Bernoulli free boundary problems. The method is based on the pseudo-transient
continuation with which a moving boundary problem is constructed and its
steady-state solution is taken as the solution of the underlying Bernoulli free
boundary problem. The moving boundary problem is solved in a split manner at
each time step: the moving boundary is updated with the Euler scheme, the
interior mesh points are moved using a moving mesh method, and the
corresponding initial-boundary value problem is solved using the linear finite
element method. The method can take full advantages of both the
pseudo-transient continuation and the moving mesh method. Particularly, it is
able to move the mesh, free of tangling, to fit the varying domain for a
variety of geometries no matter if they are convex or concave. Moreover, it is
convergent towards steady state for a broad class of free boundary problems and
initial guesses of the free boundary. Numerical examples for Bernoulli free
boundary problems with constant and non-constant Bernoulli conditions and for
nonlinear free boundary problems are presented to demonstrate the accuracy and
robustness of the method and its ability to deal with various geometries and
nonlinearities.
