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Error analysis of scalar auxiliary variable finite element methods for the Landau--Lifshitz--Bloch equation

This paper proposes and rigorously analyzes two fully discrete, linear, and unconditionally energy-stable finite element schemes based on the scalar auxiliary variable approach for solving the Landau--Lifshitz--Bloch equation in the high-temperature regime, establishing optimal-order error estimates and providing the first second-order accurate linear energy-stable method for this problem.

Original authors: Agus L. Soenjaya

Published 2026-03-04

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CC BY 4.0

Original authors: Agus L. Soenjaya

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict how a giant, invisible magnet behaves when you heat it up.

In the world of physics, magnets usually have a "Curie temperature." Below this temperature, they act like rigid, organized soldiers (ferromagnets). But once you heat them past this point, they start to wobble, lose their perfect alignment, and behave more like a chaotic, hot soup of magnetic particles. This chaotic behavior is described by a complex mathematical rule called the Landau-Lifshitz-Bloch (LLB) equation.

The problem? This equation is incredibly difficult to solve on a computer. It's like trying to predict the exact path of every single drop of water in a boiling pot while the pot itself is shaking. If your computer simulation isn't careful, it might invent fake energy out of nowhere, making the magnet behave in ways that are physically impossible (like getting hotter without a heat source).

This paper, by Agus L. Soenjaya, introduces two new, clever ways to solve this equation on a computer. Here is the breakdown using simple analogies:

1. The Problem: The "Leaky Bucket"

Think of the magnet's energy like water in a bucket. In the real world, as time passes, the water (energy) should slowly leak out as the system cools down or stabilizes. This is called energy dissipation.

Many old computer methods are like buckets with holes in the bottom that you can't see. Sometimes, the simulation accidentally adds water to the bucket, making the magnet act crazy and unstable. The goal of this paper is to build a bucket that guarantees the water level only goes down, never up, no matter how fast you pour it.

2. The Solution: The "Scalar Auxiliary Variable" (SAV)

The author uses a technique called SAV (Scalar Auxiliary Variable).

The Analogy: Imagine you are trying to balance a very wobbly, heavy stack of books (the complex magnetic field). It's hard to keep it steady.
The Trick: Instead of trying to balance the whole stack at once, you attach a small, magical counterweight (the "Scalar Auxiliary Variable") to the side. This counterweight doesn't change the physics of the books, but it acts like a "safety net" or a "thermostat" for the math.
The Result: This safety net allows the computer to use simple, straight-line calculations (linear methods) instead of complex, twisting loops. It makes the math much faster and ensures the "energy bucket" never overflows.

3. The Two New Schemes

The paper proposes two specific ways to use this safety net:

Scheme A (The Steady Walker): This uses a "semi-implicit Euler" method. Think of this as walking forward one small, careful step at a time. It's very reliable and stable, but it's a bit slow. It's accurate to the first order (like measuring distance with a ruler that has big markings).
Scheme B (The Sprinter): This uses a "BDF2" method. This is like taking two steps at once, looking ahead to where you will be, and adjusting your stride. It is twice as fast and twice as accurate as Scheme A.
- Why this matters: The author notes that no one had successfully built a "Sprinter" (second-order accurate) method for this specific hot-magnet problem that was also guaranteed to be stable. This is a world-first achievement.

4. The Proof: "The Math Detective Work"

Just because a method looks good on paper doesn't mean it works in reality. The author spent a huge part of the paper doing rigorous "error analysis."

The Analogy: Imagine you are a detective trying to prove that your new map (the computer simulation) leads to the exact same destination as the real terrain (the actual physics).
The Challenge: The map is made of tiny triangles (finite elements), and the terrain is smooth. The author had to prove that no matter how small the triangles are, the difference between the map and the terrain shrinks at a predictable, perfect rate.
The Result: The author proved that both methods are unconditionally stable. This means they won't crash or produce nonsense, even if you take huge time steps. They also proved the errors are "optimal," meaning the computer is doing the best possible job given the tools available.

5. The Experiments: "The Movie Reel"

Finally, the author ran computer simulations to show off the new methods.

They simulated a magnet heating up.
The computer showed the magnetic "spins" (tiny arrows) wobbling and eventually fading away as the heat took over.
They checked the "energy graph" and saw it smoothly slide down, just like a real magnet losing energy.
They compared the new "Sprinter" method against the "Steady Walker" and confirmed the Sprinter was indeed twice as accurate.

Summary

In short, this paper solves a long-standing headache in computational physics. It provides two new, super-stable, and highly accurate tools to simulate how magnets behave when they get hot.

Why it matters: This is crucial for technologies like Heat-Assisted Magnetic Recording (HAMR), which is the technology used in next-generation hard drives to store massive amounts of data. To build better hard drives, engineers need to know exactly how magnets behave at high temperatures, and this paper gives them a much better way to calculate it.

The author has essentially built a new, unbreakable safety harness for the mathematical models that drive our future data storage technology.

1. Problem Statement

The paper addresses the numerical solution of the Landau–Lifshitz–Bloch (LLB) equation, a fundamental model for magnetization dynamics in ferromagnetic materials at elevated temperatures (specifically near or above the Curie temperature).

Mathematical Formulation: The equation describes the evolution of the magnetization vector $\mathbf{u}: [0, T] \times D \to \mathbb{R}^3$ under an effective field $\mathbf{H}$ . The system is a vector-valued, second-order quasilinear partial differential equation (PDE):
$\partial_t \mathbf{u} = -\gamma \mathbf{u} \times \mathbf{H} + \alpha \mathbf{H}$
$\mathbf{H} = \sigma \Delta \mathbf{u} - \kappa \mu \mathbf{u} - \kappa |\mathbf{u}|^2 \mathbf{u}$
subject to Neumann boundary conditions.
Key Challenges:
- Nonlinearity: The equation contains a cross-product term ( $\mathbf{u} \times \mathbf{H}$ ) and a cubic nonlinearity ( $|\mathbf{u}|^2 \mathbf{u}$ ), making it quasilinear rather than semilinear.
- Energy Dissipation: The physical system dissipates micromagnetic energy over time. Numerical schemes must preserve this property (energy stability) to avoid non-physical solutions.
- Vector-Valued Nature: Unlike many existing Scalar Auxiliary Variable (SAV) applications to scalar gradient flows, the LLB equation involves vector-valued unknowns, preventing the direct use of standard mean value theorems in error analysis.
- Existing Limitations: Prior numerical methods were either nonlinear (requiring iterative solvers), conditionally stable, or limited to first-order temporal accuracy.

2. Methodology

The author proposes and analyzes two fully discrete, conforming Finite Element Methods (FEM) based on the Scalar Auxiliary Variable (SAV) approach. The core idea of SAV is to introduce a scalar variable $r(t)$ to handle the nonlinear potential energy term, allowing for linear, unconditionally stable schemes.

A. Reformulation

The energy functional $E[\mathbf{u}]$ is split, and a scalar variable $r(t) = \sqrt{F[\mathbf{u}]}$ is introduced, where $F[\mathbf{u}]$ represents the nonlinear part of the energy. The LLB equation is rewritten as a coupled system involving $\mathbf{u}$ , $\mathbf{H}$ , and $r(t)$ .

B. Proposed Schemes

Two time-discretization schemes are developed using linear finite elements ( $V_h \subset H^1$ ):

Scheme 1: Semi-Implicit Euler (First-Order)
- Uses a semi-implicit Euler method for time stepping.
- The cross-product term $\mathbf{u} \times \mathbf{H}$ is treated semi-implicitly (using $\mathbf{u}^{n-1}$ ).
- The nonlinear term involving $r(t)$ is linearized using the previous time step's data.
- Result: A linear system at each time step.
Scheme 2: Linearized BDF2 (Second-Order)
- Uses the second-order Backward Differentiation Formula (BDF2).
- Employs a linear extrapolation for the nonlinear terms to maintain linearity in the algebraic system.
- Result: A linear system with second-order temporal accuracy.

C. Stability and Analysis Tools

Modified Energy: Both schemes satisfy a discrete energy law involving a modified energy functional ( $\tilde{E}$ or $\hat{E}$ ) that approximates the true physical energy.
Error Decomposition: The error is split into a Ritz projection error ( $\rho$ ) and a discrete error ( $\theta$ ).
Key Technical Innovations:
- Handling the cross-product term semi-implicitly to manage the quasilinear nature.
- Utilizing discrete Gagliardo–Nirenberg inequalities and Sobolev embeddings to bound $L^\infty$ norms of the discrete solutions.
- Proving unconditional stability (stability independent of the time step size $k$ ) with respect to the energy norm.

3. Key Contributions

First Rigorous Error Analysis for Quasilinear SAV: To the author's knowledge, this is the first rigorous error analysis of a fully discrete SAV-based method applied to a quasilinear vector-valued problem. Previous SAV analyses were largely restricted to semilinear scalar gradient flows.
Second-Order Accuracy: The paper introduces the first linear, energy-stable scheme for the LLB equation in the high-temperature regime that achieves second-order temporal accuracy (via the BDF2 approach).
Optimal Convergence Rates: The author derives optimal-order error estimates in $L^2$ $L^{2}$ , $H^1$ $H^{1}$ , and $L^\infty$ $L^{\infty}$ norms under suitable regularity assumptions.
- Scheme 1: $O(h^2 + k)$ in $H^1$ .
- Scheme 2: $O(h^2 + k^2)$ in $H^1$ .
Unconditional Energy Stability: Both schemes are proven to be unconditionally stable, ensuring the discrete energy dissipates over time, which is crucial for long-time simulations.

4. Main Results

Theoretical Proofs:
- Well-posedness: Existence and uniqueness of the discrete solution for both schemes are proven.
- Stability: Unconditional energy stability is established for both schemes.
- Convergence: Optimal error estimates are derived. For the BDF2 scheme, the error in the $L^2$ and $H^1$ norms is $O(h^2 + k^2)$ . In 2D, an $L^\infty$ estimate of $O((h^2 + k^2)|\ln h|^{1/2})$ is obtained.
Numerical Experiments:
- Simulations were performed using the FEniCS package on a 2D domain.
- Convergence Verification: Numerical results confirmed the theoretical convergence rates. Scheme 1 showed first-order temporal convergence, and Scheme 2 showed second-order temporal convergence.
- Energy Behavior: Plots of energy vs. time demonstrated the decay of both the actual and modified energies, validating the energy-stability property.
- Physical Phenomena: The simulations successfully captured physical features such as domain wall formation and the decay of magnetization magnitude to zero above the Curie temperature.

5. Significance

This work bridges a critical gap in computational micromagnetics. By providing a linear, unconditionally stable, and high-order accurate method for the LLB equation, it enables efficient and reliable simulations of thermally induced magnetization dynamics (e.g., in Heat-Assisted Magnetic Recording - HAMR). The rigorous error analysis for a quasilinear vector system sets a new standard for the application of the SAV methodology to complex, non-gradient flow PDEs, overcoming the difficulties posed by cross-product terms and vector-valued unknowns.