Quasi-optimal error estimates for the approximation of stable harmonic maps

2022/09/24 by Sören Bartels, Bartels, Sören, Christian Palus +3
Computer Science · Engineering · Mathematics · #35J62 (35J50 35J57 65N30) #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2209.11985

openalex publication_date 2022/09/24 · openalex created_date 2022/09/28 · openalex updated_date 2026/07/28

Abstract

Based on a quantitative version of the inverse function theorem and an appropriate saddle-point formulation we derive a quasi-optimal error estimate for the finite element approximation of harmonic maps into spheres with a nodal discretization of the unit-length constraint. The estimate holds under natural regularity requirements and appropriate geometric stability conditions on solutions. Extensions to other target manifolds including boundaries of ellipsoids are discussed.

Related

• Benchmarking Numerical Algorithms for Harmonic Maps into the Sphere
• Error analysis for the numerical approximation of the harmonic map heat flow with nodal constraints
• L²-Discretization Error Bounds for Maps into Riemannian Manifolds
• Projection-Based Finite Elements for Nonlinear Function Spaces
• Quadratic constraint consistency in the projection-free approximation of harmonic maps and bending isometries