Conjugate Harmonic Maps

Konrad Polthier

Abstract

We consider discrete harmonic maps that are conforming or non-conforming piecewise linear maps, and derive a bijection between the minimizers of the two corresponding Dirichlet problems. Pairs of harmonic maps with a conforming and a non-conforming component solve the discrete Cauchy-Riemann equations, and have vanishing discrete conformal energy.

As an application, the results of this work provide a thorough understanding of the conjugation algorithms of Pinkall/Polthier and Oberknapp/Polthier used in the computation of discrete minimal and constant mean curvature surfaces.

Full paper: Preprint (.pdf 380 KB, .ps.gz 375 KB)

Introduction

Discrete harmonic maps have been well studied as a basic model problem in finite element theory, while the definition of the conjugate of a discrete harmonic map was not completely settled. Here we are interested in pairs of discrete harmonic maps on a Riemann surface M which are both minimizers of the Dirichlet energy

E(u) = 1 2 � � M  | �u| 2dx,

and are solutions of the Cauchy Riemann equations

dv = *du.

We note that generically such pairs neither exist in the space of piecewise linear conforming Lagrange finite elements S_h^., nor in the space of piecewise linear non-conforming Crouzeix-Raviart elements S_h^*.

In the present paper we solve the conjugation of harmonic maps by simultaneously considering harmonic maps in S_h and S_h^* from M to R. The main result is

Theorem: Let T_h be a triangulation of a domain on a Riemann surface M in Rⁿ .

1. Let u � S_h be a minimizer of the Dirichlet energy in S_h. Then its conjugate map u^* is in S_h^* and is discrete harmonic.
2. Let v � S_h^* be a minimizer of the Dirichlet energy in S_h^*. Then its conjugate map v^* is in S_h and is discrete harmonic.
3. Let u � S_h, respectively S_h^* be discrete harmonic in S_h, respectively S_h^*. Then u^** = -u.

Our interest in harmonic maps arose from the study of numerical algorithms to compute the conjugate of minimal and constant mean curvature surfaces in Euclidean three-space. In the algorithms and, the conjugate of a minimal surface is obtained via the conjugate of discrete harmonic maps. Conjugate harmonic maps were defined on the dual graph of the edge graph of the original minimal surface. Although these methods were successful and allowed the numerical computation of a number of minimal surfaces for the first time, they provided no further hints on the harmonicity properties of the conjugate harmonic maps. The results of the present paper provide a thorough understanding of the geometric conjugation algorithms in Pinkall and Polthier and in Oberknapp and Polthier by relating the geometric discretization techniques to the context of finite element methods, and the convergence of the conjugation of minimal surfaces.

Convergence of conforming harmonic maps has been shown by Tsuchiya . As a more general result for surfaces, Dziuk and Hutchinson obtained optimal convergence results in the H¹ norm for the finite element procedure of the Dirichlet problem of surfaces with prescribed mean curvature have been obtained by . Compare M�ller, Struwe, and Sver�k for harmonic maps on planar lattices using the five-point Laplacian.