MINIMAL-SURFACES AND SOBOLEV GRADIENTS

We treat the problem of computing triangle-based piecewise linear approximations to parametric minimal surfaces in R(3). More specifically, given a triangulation T of the unit square Omega and a function f(0) from the nodes of T into R(3), We Seek a function f from the nodes of T into R(3) such that f agrees with f(0) on the boundary of Omega, and the triangulated surface area corresponding to the image of f is minimal. We employ a descent method in which, at each iteration, the gradient of the surface area functional is computed with respect to an inner product that depends on the current approximation to f. Test results: show that, starting with extremely poor initial estimates, a few descent iterations produce approximations in the vicinity of the solution. We also introduce a new characterization of minimal surfaces that eliminates the potential problem of triangle areas approaching zero. In place of the surface area functional, we minimize a functional whose critical points are uniformly parameterized minimal surfaces. This not only results in rapid convergence of the descent method, but also simplifies the expressions for gradients and Hessians.