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Finite-Element Discretization of Static Hamilton-Jacobi Equations Based on a Local Variational Principle

Folkmar Bornemann, Christian Rasch

TL;DR

The paper develops a linear finite-element discretization for static Hamilton–Jacobi equations on unstructured meshes by solving local Dirichlet problems with a local Hopf--Lax variational principle, yielding a simple, monotone convergence theory. The method defines a Hopf--Lax update $\Lambda_h$ and solves the nonlinear system $u_h=\Lambda_h u_h$ with boundary data; convergence to the viscosity solution is established via a consistency framework using a modified Hamiltonian $\tilde{H}$ and Arzelà–Ascoli. An explicit 2D Hopf--Lax update for generalized eikonal equations with metric $M(x)$ is provided, along with efficient computation strategies. The paper further introduces an adaptive nonlinear Gauss--Seidel solver, which significantly accelerates convergence and demonstrates strong performance gains in numerical experiments compared to standard Gauss--Seidel and the ordered upwind method (OUM).

Abstract

We propose a linear finite-element discretization of Dirichlet problems for static Hamilton-Jacobi equations on unstructured triangulations. The discretization is based on simplified localized Dirichlet problems that are solved by a local variational principle. It generalizes several approaches known in the literature and allows for a simple and transparent convergence theory. In this paper the resulting system of nonlinear equations is solved by an adaptive Gauss-Seidel iteration that is easily implemented and quite effective as a couple of numerical experiments show.

Finite-Element Discretization of Static Hamilton-Jacobi Equations Based on a Local Variational Principle

TL;DR

The paper develops a linear finite-element discretization for static Hamilton–Jacobi equations on unstructured meshes by solving local Dirichlet problems with a local Hopf--Lax variational principle, yielding a simple, monotone convergence theory. The method defines a Hopf--Lax update and solves the nonlinear system with boundary data; convergence to the viscosity solution is established via a consistency framework using a modified Hamiltonian and Arzelà–Ascoli. An explicit 2D Hopf--Lax update for generalized eikonal equations with metric is provided, along with efficient computation strategies. The paper further introduces an adaptive nonlinear Gauss--Seidel solver, which significantly accelerates convergence and demonstrates strong performance gains in numerical experiments compared to standard Gauss--Seidel and the ordered upwind method (OUM).

Abstract

We propose a linear finite-element discretization of Dirichlet problems for static Hamilton-Jacobi equations on unstructured triangulations. The discretization is based on simplified localized Dirichlet problems that are solved by a local variational principle. It generalizes several approaches known in the literature and allows for a simple and transparent convergence theory. In this paper the resulting system of nonlinear equations is solved by an adaptive Gauss-Seidel iteration that is easily implemented and quite effective as a couple of numerical experiments show.

Paper Structure

This paper contains 21 sections, 13 theorems, 90 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Existence and Uniqueness of Viscosity Solutions
  3. The Support Function of the Zero-Level Set
  4. Example.
  5. The Finite-Element Discretization
  6. Linear Finite Elements.
  7. The Idea.
  8. The Discretization
  9. Remark.
  10. Existence and Uniqueness of the Finite-Element Solution
  11. Step 1.
  12. Step 2.
  13. Step 3.
  14. Convergence of the Finite-Element Discretization
  15. Remark
  16. ...and 6 more sections

Key Result

Theorem 1

Assume (H1)--(H4). The Dirichlet problem (eq.Dirich) has a viscosity solution $u$ if and only if the boundary condition satisfies the compatibility condition (H5). A specific viscosity solution is then given by the Hopf--Lax formula

Figures (4)

  • Figure 1: The neighborhood $\omega_h(x_h)$ of $x_h \in \Omega_h$, that is, the collection of all simplices (like the one shaded in dark) that have $x_h$ as a vertex.
  • Figure 2: Geometry of the minimization of $\cos(\delta)\|y-y_h\| + \|x_h-y\|$ for $y \in [y_h,z_h]$. Note that for $\delta > \pi/2$ the segment through $y$ perpendicular to $l_\delta$ has negative length $\cos(\delta)\cdot \|y-y_h\|$.
  • Figure 3: Left: Contour plot of the distance function over the parameter plane. Right: Accuracy/complexity of adaptive Gauss-Seidel iteration in comparison to the Gauss-Seidel iteration and OUM.
  • Figure 4: Left: Value function of some min-time optimal control problem. Right: Complexity/accuracy of the methods in comparison.

Theorems & Definitions (24)

  • Theorem 1: P.-L. Lions 0497.35001
  • Theorem 2: H. Ishii 0644.35017
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Theorem 6
  • proof
  • ...and 14 more