🌀 nonlinear sciences

An adaptive finite element method for the infinity Laplacian

This paper presents an adaptive finite element method for the infinity Laplacian that utilizes residual-based a posteriori error estimators to effectively handle solution singularities and restore optimal convergence rates.

Original authors: Omar Lakkis, Tristan Pryer

Published 2026-06-04

📖 5 min read🧠 Deep dive

CC BY 3.0

Original authors: Omar Lakkis, Tristan Pryer

Original paper licensed under CC BY 3.0 (http://creativecommons.org/licenses/by/3.0/). ✨ 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 stretch a rubber sheet over a complex frame to make it as flat as possible, but with a very strange rule: the sheet must minimize the steepest slope anywhere on its surface. This is the essence of the Infinity Laplacian, a mathematical problem that describes how things behave when they are trying to be as "smooth" as possible in the most extreme sense.

The problem is tricky because the solution often has sharp corners or "singularities" (like a pyramid peak or a sharp ridge) where the math gets messy and breaks down. Traditional computer methods struggle with these sharp points.

Here is what the authors, Omar Lakkis and Tristan Pryer, did to solve it, explained simply:

1. The Problem: A Math Puzzle with Sharp Edges

Think of the Infinity Laplacian as a rule for finding the "perfect" shape for a surface. If you try to solve this using standard computer techniques (like dividing the surface into a grid of tiny squares), the sharp corners cause the computer to get confused or produce inaccurate results. It's like trying to measure the slope of a jagged mountain peak with a ruler meant for rolling hills.

2. The Solution: A New Way to "Feel" the Shape

The authors created a new Finite Element Method (FEM). In simple terms, this is a way for a computer to break a complex shape into small, manageable pieces (like a mosaic) to solve the equation.

However, because the math involves a "second derivative" (which measures how the slope changes, or the curvature), and the solution is so sharp, the computer can't just calculate it directly.

The Trick: They introduced a "helper variable." Imagine trying to figure out the curvature of a bumpy road. Instead of trying to measure the curve directly, they asked the computer to guess the curve, check how wrong the guess was, and then adjust. They treated the "curvature" as a separate character in the story that helps solve the main equation.

3. The "Relaxation" Technique: Taking Small Steps

The math behind this problem is "degenerate," meaning it can get stuck or become unstable. To fix this, the authors used a technique called Laplacian relaxation.

The Analogy: Imagine you are trying to walk to a specific spot on a foggy hill. Instead of trying to jump straight to the destination (which might make you slip), you take small, cautious steps. You look at where you are, take a step, check your balance, and then take another.
In their math, they added a "time step" (called $\tau$ ). They pretend the solution is evolving slowly over time, taking small steps until it settles into the final, perfect shape. This prevents the computer from crashing when it hits those sharp corners.

4. The Adaptive Algorithm: Zooming In Where It Hurts

This is the most important part of their paper. They realized that if they used the same size of "tiles" (mesh) everywhere, they would waste computing power on smooth areas and still miss the details in the sharp areas.

So, they built an adaptive system:

The Metaphor: Imagine you are looking at a map. If you are looking at a flat ocean, you don't need a high-resolution satellite view. But if you are looking at a jagged coastline, you zoom in immediately to see the details.
How it works: The computer calculates an "error estimator" (a score that tells it how confused it is). If the score is high (meaning the math is messy or the solution is sharp), the computer automatically cuts those tiles into smaller, finer pieces. If the score is low (the area is smooth), it leaves the tiles big.
The Result: This allows the computer to focus its energy exactly where the "sharp corners" are, regaining the best possible speed and accuracy.

5. The Results: Proving It Works

The authors tested their method on two types of problems:

A Smooth Problem: They tested it on a shape they already knew the answer to. The computer got the answer perfectly, with the expected high speed of convergence.
A Sharp Problem: They tested it on a shape with a known "singularity" (a sharp point, specifically the Aronsson solution). Even though the math is very difficult here, their adaptive method still worked. It didn't get the perfect theoretical speed because of the sharpness, but it was much better than non-adaptive methods and proved that the "zooming in" strategy works.

Summary

In short, the authors built a smart computer program that solves a very difficult math problem about "extreme smoothness." Instead of trying to force a rigid grid to fit a jagged shape, they created a system that:

Uses a helper variable to handle the complex math.
Takes small, safe steps to avoid instability.
Automatically zooms in on the messy, sharp parts of the solution to get the best possible answer without wasting time on the smooth parts.

They showed that this approach successfully recovers the best possible accuracy, even when the solution has sharp, singular edges.

Technical Summary: An Adaptive Finite Element Method for the Infinity Laplacian

Problem Statement
The paper addresses the numerical approximation of the infinity Laplacian, a nonlinear partial differential equation (PDE) defined as $\Delta_\infty u := (\nabla u \otimes \nabla u) : D^2 u / |\nabla u|^2$ . This operator arises as the formal limit of $p$ -harmonic functions as $p \to \infty$ and is associated with the problem of finding absolutely minimizing Lipschitz extensions. The primary challenge in solving the infinity Laplacian is its non-divergence structure and the fact that solutions are often singular (lacking $C^2$ regularity), rendering classical numerical schemes difficult to formulate. While viscosity solutions provide a rigorous existence and uniqueness framework, standard finite element methods (FEM) struggle with the non-variational nature of the equation. Previous numerical approaches have been limited to finite difference methods or the "vanishing moment method," which approximates the second-order operator via fourth-order biharmonic equations.

Methodology
The authors propose a non-variational finite element method that avoids the complications of fourth-order operators. The core methodology involves three main components:

Linearization and Relaxation: The authors employ a fixed-point linearization strategy. To handle the degeneracy of the problem, they introduce a Laplacian relaxation technique, transforming the steady-state problem into the limit of an evolution equation: $\partial_t(\Delta u) + \Delta_\infty u = f$ . This results in a sequence of linear non-divergence PDEs where a time-step parameter $\tau$ is introduced to ensure stability and convergence to the steady state.
Finite Element Formulation: The method utilizes a non-variational Galerkin procedure. A key innovation is the treatment of the Hessian $D^2 u$ as an auxiliary variable. The authors define a "finite element Hessian," $H[U]$ , within a discontinuous polynomial space ( $V_D$ ) via a weak formulation involving jumps and averages across element edges. This allows the method to handle the non-divergence structure without requiring $C^1$ continuity.
Adaptive Algorithm: Recognizing that solutions are often singular, the authors conduct an a posteriori error analysis. They derive residual-based error estimators (both interior residuals and jump residuals) to drive an adaptive mesh refinement algorithm. This approach aims to recover optimal convergence rates by refining the mesh specifically in regions of high error or singularity.

Key Contributions

Novel FEM Scheme: The paper presents a finite element method for the infinity Laplacian that does not rely on approximating a fourth-order operator, distinguishing it from the vanishing moment method.
A Posteriori Analysis: The authors derive a rigorous a posteriori error bound for the discrete problem, providing a theoretical basis for adaptive refinement.
Handling Degeneracy: The introduction of the Laplacian relaxation parameter $\tau$ offers a practical mechanism to circumvent the degeneracy of the infinity Laplacian during the linearization process.

Numerical Results
The method is tested using the FEniCS framework on the domain $\Omega = [-1, 1]^2$ with linear continuous elements for the solution and piecewise constant discontinuous elements for the Hessian.

Classical Solutions: For problems with smooth solutions (e.g., $u = |x|^2$ ), the method demonstrates optimal convergence rates: $O(h^2)$ in the $L^2$ norm and $O(h)$ in the $H^1$ norm.
Singular Solutions: For the homogeneous problem with a known singular viscosity solution (the Aronsson solution $u = |x|^{4/3} - |y|^{4/3}$ ), uniform mesh refinement yields suboptimal convergence rates ( $O(h^{1.8})$ in $L^2$ and $O(h^{0.8})$ in $H^1$ ) due to the singularity.
Adaptive Performance: The paper notes that the adaptive procedure, driven by the derived error estimators, successfully regains optimal convergence rates for singular solutions, effectively mitigating the impact of the lack of regularity.

Significance and Claims
The paper claims to provide a robust numerical framework for the infinity Laplacian that is computationally efficient and theoretically grounded. By utilizing a non-variational formulation with an auxiliary Hessian variable and an adaptive strategy, the authors demonstrate that it is possible to achieve optimal convergence rates even for solutions that are not classically smooth. The work bridges the gap between the theoretical existence of viscosity solutions and their practical numerical approximation, offering an alternative to finite difference and higher-order approximation methods. The authors emphasize that their adaptive procedure is essential for handling the singular nature of infinity harmonic functions, as evidenced by the recovery of optimal rates in numerical experiments.