  We present a numerical method for solving Weyl's embedding problem which
consists of finding a global isometric embedding of a positively curved and
positive-definite spherical 2-metric into the Euclidean three space. The method
is based on a construction introduced by Weingarten and was used in Nirenberg's
proof of Weyl's conjecture. The target embedding results as the endpoint of an
embedding flow in R^3 beginning at the unit sphere's embedding. We employ
spectral methods to handle functions on the surface and to solve various
(non)-linear elliptic PDEs. Possible applications in 3+1 numerical relativity
range from quasi-local mass and momentum measures to coarse-graining in
inhomogeneous cosmological models.
