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Quadratic constraint consistency in the projection-free approximation of harmonic maps and bending isometries

Georgios Akrivis, Sören Bartels, Christian Palus

TL;DR

The paper addresses numerical approximation of harmonic maps and bending isometries under holonomic unit-length constraints without projection steps. It introduces a projection-free scheme based on a BDF2 time discretization and analyzes energy stability and constraint violation, establishing unconditional energy decay and a linear bound on constraint violation with a potential quadratic rate under mild discrete regularity. The main finding is that the BDF2-based method delivers second-order accuracy in constraint violation and robust stability, outperforming backward Euler approaches in many tests. Experiments on harmonic maps and bending isometries validate the approach, showing faster constraint decay and lower energies, demonstrating the method's effectiveness for constrained PDEs in practice.

Abstract

We devise a projection-free iterative scheme for the approximation of harmonic maps that provides a second-order accuracy of the constraint violation and is unconditionally energy stable. A corresponding error estimate is valid under a mild but necessary discrete regularity condition. The method is based on the application of a BDF2 scheme and the considered problem serves as a model for partial differential equations with holonomic constraint. The performance of the method is illustrated via the computation of stationary harmonic maps and bending isometries.

Quadratic constraint consistency in the projection-free approximation of harmonic maps and bending isometries

TL;DR

The paper addresses numerical approximation of harmonic maps and bending isometries under holonomic unit-length constraints without projection steps. It introduces a projection-free scheme based on a BDF2 time discretization and analyzes energy stability and constraint violation, establishing unconditional energy decay and a linear bound on constraint violation with a potential quadratic rate under mild discrete regularity. The main finding is that the BDF2-based method delivers second-order accuracy in constraint violation and robust stability, outperforming backward Euler approaches in many tests. Experiments on harmonic maps and bending isometries validate the approach, showing faster constraint decay and lower energies, demonstrating the method's effectiveness for constrained PDEs in practice.

Abstract

We devise a projection-free iterative scheme for the approximation of harmonic maps that provides a second-order accuracy of the constraint violation and is unconditionally energy stable. A corresponding error estimate is valid under a mild but necessary discrete regularity condition. The method is based on the application of a BDF2 scheme and the considered problem serves as a model for partial differential equations with holonomic constraint. The performance of the method is illustrated via the computation of stationary harmonic maps and bending isometries.
Paper Structure (8 sections, 6 theorems, 72 equations, 3 figures, 3 tables, 2 algorithms)

This paper contains 8 sections, 6 theorems, 72 equations, 3 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Auxiliary results
  3. Discrete time derivatives
  4. BDF-adapted norm
  5. Main result
  6. Experiments
  7. Harmonic maps
  8. Bending isometries

Key Result

Lemma 2.1

For every sequence $(u^n)$ and $N\ge 2$ we have for the seminorms that $c_{12}^{-1} |(u^n)|_{\tau,1} \le |(u^n)|_{\tau,2} \le c_{12} |(u^n)|_{\tau,1}$ with $c_{12}\ge 1$.

Figures (3)

  • Figure 1: Nodal interpolant of the inverse stereographic projection $\pi_{\rm st}^{-1}$ in Example \ref{['ex:invstereo']} on a coarse grid (left and middle) and perturbed initial configuration $u_h^0$ (right).
  • Figure 2: Nodal values of the rough initial data $u_h^0$ for the approximation of Example \ref{['ex:invstereo']} on a coarse grid (left) and final iterate for the BDF2 method with $\tau=2^{-8}$ (middle, right).
  • Figure 3: Evolution of an initially flat Möbius strip from Example \ref{['ex:moebius']} using the implicit Euler (top row) and BDF2 methods (bottom row) realizing discrete $H^2$-gradient flows with step sizes $\tau=2^{-7}$. The coloring represents the constraint violation.

Theorems & Definitions (17)

  • Lemma 2.1: Norm equivalence
  • proof
  • Lemma 2.2: Inverse estimate
  • proof
  • Lemma 2.3: Discrete chain rule
  • proof
  • Remark 2.4
  • Proposition 3.1: Initialization
  • proof
  • Proposition 3.2: Energy decay
  • ...and 7 more