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

TL;DR

This paper introduces a projection-free iterative method for approximating harmonic maps with second-order accuracy and unconditional energy stability, applicable to PDEs with holonomic constraints, demonstrated through stationary harmonic maps and bending isometries.

Contribution

It presents a novel BDF2-based scheme that achieves second-order constraint accuracy and energy stability without projections, advancing numerical methods for constrained PDEs.

Findings

Method achieves second-order accuracy in constraint violation.

Unconditionally energy stable scheme demonstrated.

Effective in computing harmonic maps and bending isometries.

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.