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Minimal surfaces and Sobolev gradients. (English) Zbl 0857.35004
The authors treat the problem of computing triangle-based piecewise linear approximations to parametric minimal surfaces in the Euclidean 3-space. They employ the Sobolev metric method to descend the surface-area functional at each iteration. Test results show that, starting with extremely poor initial estimates, a few descent iterations produce approximations in the vicinity of the solution. They also introduce a new characterization of minimal surfaces that eliminates the potential problem of triangle area approaching zero. In place of the surface area functional, they minimize a functional whose critical points are uniformly parametrized minimal surfaces. This leads to both rapid convergence of the descent method and simplifying the expressions for gradients and Hessians.

35A15 Variational methods applied to PDEs
65N06 Finite difference methods for boundary value problems involving PDEs
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
65N99 Numerical methods for partial differential equations, boundary value problems
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