  Let $N\ge3$ be an integer and $B$ be a smooth, compact, oriented,
$(N-2)$-dimensional boundary in ${\Bbb R}^{N}$. In 1960, H. Federer and W.
Fleming proved that there is an $(N-1)$-dimensional integral current spanning
surface of least area. The proof was by compactness methods and
non-constructive. In 1970 H. Federer proved the definitive regularity result
for such a codimension one minimizing surface. Thus it is a question of long
standing whether there is a numerical algorithm that will closely approximate
the area minimizing surface. The principal result of this paper is an algorithm
that solves this problem.
  Specifically, given a neighborhood $U$ around $B$ in ${\Bbb R}^{N}$ and a
tolerance $\epsilon>0$, we prove that one can explicitly compute in finite time
an $(N-1)$-dimensional integral current $T$ with the following approximation
requirements:
  (1) spt$(\partial T)\subset U$.
  (2) $B$ and $\partial T$ are within distance $\epsilon$ in the Hausdorff
distance.
  (3) $B$ and $\partial T$ are within distance $\epsilon$ in the flat norm
distance.
  (4) ${\mathbb M}(T)<\epsilon+\inf\{{\mathbb M}(S):\partial S=B\}$.
  (5) Every area minimizing current $R$ with $\partial R=\partial T$ is within
flat norm distance $\epsilon$ of $T$.
