The discrete Douglas problem: convergence results

In Pozzi (2004a) we defined a general framework to solve the problem of finding and justifying an optimal fully discrete finite-element procedure for approximating annulus-like minimal surfaces. Here we prove convergence estimates in various norms. The main results proved in this paper can be informally stated as follows. Let Γ1, Γ2 ∈ Rn be two disjoint closed Jordan curves, rectifiable and with given orientation and set Γ = (Γ1,Γ2). Let Cλ be a cylinder of unit radius and length λ ∈ (0,∞). The formulation of annulus-like minimal surfaces which we use is the following (for motivation and other equivalent formulations we refer the reader to Pozzi, 2004a). Let F be the class of maps u : Cλ → Rn , for all possible choices of λ > 0, such that u|∂Cλ : ∂Cλ → Γ is monotonic and u is harmonic. The function u ∈ F defined on Cλ is said to be a minimal surface if u is stationary in F for the Dirichlet energy D(u) = 2 ∫ Cλ |∇u|2. Such a map u provides a harmonic and conformal parametrization of the corresponding minimal surface. The numerical method can be described as follows. For any λ > 0, let Gλh be a quasi-uniform triangulation of Cλ controlled by h. We can consider Gλh as a one-parameter family of triangulations corresponding to the one-parameter family of domains Cλ. Let Fh be the class of continuous piecewise linear maps uh : Cλ → Rn , for all possible choices of λ > 0, which are discrete harmonic and for which uh(φ j ) ∈ Γ whenever φ j is a boundary node of Cλ. Note that we do not require the monotonicity of uh |∂Cλ . A function uh ∈ Fh defined on Cλh is said to be a discrete minimal surface if uh is stationary within Fh for the Dirichlet energy D(uh) = 2 ∫ Cλh |∇uh |2. A member of Fh is determined by its values at the boundary nodes and by the knowledge of the length λh of its domain. The first main result is that if u : Cλ → Rn is a ‘non-degenerate’, harmonic and conformally parametrized minimal surface spanning Γ , then there exist λh ∈ (0,∞) and a discrete minimal surface uh : Cλh → Rn such that if we denote by σμ the cylinder transformation of the form σμ : C1 → Cμ, †Email: [email protected]