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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.

MSC:
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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