Unconditional Optimal Error Estimates and Energy Stability for a Linearly Implicit Mass-Lumped Projection Finite Element Method for the Harmonic Map Flow

Jul 16, 2026

A linearly implicit mass-lumped finite element method is proposed for the heat flow of harmonic maps into the unit sphere. The scheme applies a linear predictor followed by a nodal projection that preserves the unit-length constraint exactly at finite element nodes. Using discrete inner products consistently in mass and stiffness terms, the method yields nodal orthogonality and a contraction property for the projected error in the discrete L2 norm. On tensor-product Cartesian meshes, it guarantees nonexpansive discrete Dirichlet energy and an unconditional discrete energy dissipation law. For sufficiently smooth solutions, optimal error bounds are proved without coupling constraints between time step and mesh size.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages31

Reading Time1 h 18 min

Abstract

Introduction

Heat flow of harmonic maps and energy identity

Prior work on unconditional convergence and constraint preservation

Discrete constraint enforcement methods

arXiv :2607 .02179v1 [math .NA] 2 Jul 2026 UNCONDITIONAL OPTIMAL ERROR ESTIMATES AND ENERGY STABILITY FOR ALINEARLY IMPLICIT MASS-LUMPED PROJECTION FINITE ELEMENT METHOD FOR THE HARMONIC MAP FLOW YONGYONG CAI∗ AND XINGWEI YANG† Abstract. We propose and analyze a linearly implicit mass-lumped finite element method for the heat flow of harmonic maps into the unit sphere. The method consists of a linear predictor followed by a nodal projection and therefore preserves the unit-length constraint exactly at all finite element nodes. The predictor is derived from a cross-product reformulation of the equation and is shown to be equivalent to a mass-lumped discretization of the original formulation with a correction term enforcing nodal orthogonality, as well as to a tangent plane scheme. A key ingredient is the consistent use of the discrete inner product in both the mass and stiffness terms. This yields a nodal orthogonality relation implying that the auxiliary solution lies on or outside the unit sphere at every node. Consequently, the projection is well defined and the projected error satisfies a contraction property in the discrete L2-norm. On Cartesian rectangular and cuboidal tensor-product meshes, the nodal projection is also nonexpansive in a discrete Dirichlet energy, which gives an unconditional discrete energy dissipation law. For sufficiently smooth solutions, we prove optimal error estimates without any coupling condition between the time step and the mesh size: the method converges with order O(∆t+h2 ) in ℓ∞ (0, T; L2 ) and order O(∆t+h) in ℓ2 (0, T; H1 ) . The proof combines the projected-error contraction, quadrature consistency estimates, edge-based cancellation identities, and a bootstrap argument for controlling nonlinear terms. Numerical experiments confirm the predicted convergence rates and the discrete energy decay. Key words. heat flow of harmonic maps, finite element methods, unconditional convergence, mass lumping MSC codes. 65M60, 65M12, 65M15, 35K55, 58E20 1. Introduction. For a bounded Lipschitz domain Ω ⊂ Rd with d ∈ {1, 2 , 3} and given T > 0, the heat flow of harmonic maps reads as (1 . 1) ∂tm = ∆m + |∇m|2 m in Ω × (0, T], (1.2) ∂nm = 0 on ∂Ω × (0, T], (1 .3) m = m0 in Ω × {0}, where the initial data m0 ∈ H 1 (Ω) satisfies |m0 (x)| = 1 for almost every x ∈ Ω . The symbol | · | denotes the Euclidean norm for a vector and the Frobenius norm for a matrix, ∇ is the gradient operator, ∆ is the Laplacian operator, n is the outward unit normal vector on the boundary ∂Ω . Moreover, ∂tm and ∂nm denote the time derivative and the normal derivative of m, respectively. Denoting by S 2 the unit sphere in R3 , the solution to (1 . 1)–(1 .3) intrinsically belongs to S2 , i.e. , m (x, t) satisfies the pointwise constraint (1.4) |m(x, t)| = 1 in Ω × (0, T] . It also satisfies the following energy identity for a.e. t ∈ (0, T]: E (m(x, t)) + Z0 t ∥m(x, s) × ∆m(x, s)∥2L2 ds = E (m0 ) , E (m) := 12 ZΩ |∇m|2 dx. The relevant results on local existence, uniqueness, and finite-time blow-up behavior for the system (1.1)–(1.3) can be found in [20, 21] . As a fundamental equation in many models whose exact solution inherently satisfies a unit-length constraint, the heat flow of harmonic maps has natural applications in various physical scenarios, e.g., the Ericksen–Leslie model for nematic liquid crystal flow [16], the Landau–Lifshitz equation for magnetization dynamics [33, 28], color image denoising [42], and mean curvature flow [32] . For the numerical approximation of the evolution problem (1.1)–(1.3) and the related Landau-Lifshitz equation, optimal error estimates of finite element methods (FEMs) have been studied recently. Gao established the optimal unconditional convergence result for the Landau-Lifshitz equation in [27] by a new linearization and the error splitting technique developed in [35] . Akrivis et al. [2] derived optimal-order ∗ Laboratory of Mathematics and Complex Systems and

FAQs

How does the method preserve the unit-length constraint at discrete nodes?

It uses a linear predictor followed by a nodal projection. The projection is constructed so that the unit-length constraint is satisfied exactly at every finite element node.

What ensures stability in the discrete setting?

Consistent use of the discrete inner product in both mass and stiffness terms yields a nodal orthogonality relation and implies a contraction property for the projected error. On tensor-product Cartesian meshes, it also leads to nonexpansiveness in a discrete Dirichlet energy, giving an unconditional discrete energy dissipation law.

What are the proven convergence rates and do they require coupling between time step and mesh size?

For sufficiently smooth solutions, the method converges with order O(Δt + h^2) in ℓ∞(0,T;L2) and order O(Δt + h) in ℓ2(0,T;H1). The analysis proves optimal error estimates without any coupling condition between the time step and the mesh size.

Jul 16, 2026

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