High order numerical approximation of minimal surfaces

Abstract

We present an algorithm for finding high order numerical approximations of minimal surfaces with a fixed boundary. The algorithm employs parametrization by high order polynomials and a linearization of the weak formulation of the Laplace-Beltrami operator to arrive at an iterative procedure to evolve from a given initial surface to the final minimal surface. For the steady state solution we measure the approximation error in a case where the exact solution is known (the catenoid). In the framework of parametric interpolation, the choice of interpolation points (mesh nodes) is directly affecting the approximation error, and we discuss how to best update the mesh on the evolutionary surface such that the parametrization remains smooth. In our test case we achieve exponential convergence in the approximation of the minimal surface as the polynomial degree increases, but the rate of convergence greatly differs with different choices of mesh update algorithms.

An algorithm for prescribed mean curvature using isogeometric methods

“…In our simulations we have used an L 2 projection. Also for the implementation of this algorithm, it is worth observing the simplified form that (10) takes when considering only normal perturbations.…”

“…Their intricate approach was necessary to circumvent the lack of regularity of the involved spaces, restricted to surfaces that are homeomorphic to a disc. Recently, an algorithm using high-order polynomials has been proposed by Tråsdal and Rønquist [10]. Extensions for the case of prescribed mean curvatures appear in [5,6].…”

Numerical solution of nonlinear elliptic equation with nonlocal condition

“…When function H(x, y, u) is defined as in equation (11) and the domain Ω is supposed to be the circle of radius R, then exact value of λ of problem (1)-(3) belongs to interval [−(γR/σ) sin α, (γR/σ) sin α]. So, λ 0 h could be chosen from this particular interval.…”

“…The first papers on the methods of solution of the minimal surface equation are rather early (see [5,6] and references therein). Anyway, interest in this area of numerical analysis is still strong [7][8][9][10][11].…”

The Plateau-Bézier Problem with Weak-Area Functional

“…The minimal surface has attracted scientists for many years and has been studied extensively in many literatures, such as [5,12,13,[21][22][23]. Part of the interest stems from the fact that it is so easily realizable physically in the form of soap films, and for this reason it has been studied not only mathematically, but also physically for many years [19]. The fascinating characters of minimal surface make it to be widely used in many areas such as architecture, material science, ship manufacture, biology and so on [17].…”