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A high-order local discontinuous Galerkin method for the $p$-Laplace equation

Yue Wu, Yan Xu

TL;DR

This work develops a high-order local discontinuous Galerkin method for the nonlinear $p$-Laplace equation by reformulating the discretization as a convex minimization problem and solving it with a weighted preconditioned gradient descent to achieve $hk$-independent convergence. A discrete energy functional $J_h$ is minimized on a broken Sobolev space, with a DG gradient operator $D_{DG}$ used to define the primal problem and gradient variables. The authors establish solvability, equivalence between the LDG weak form and the minimization form, and provide a priori error estimates in a mesh-dependent energy norm under sufficient regularity, revealing the potential for high-order accuracy. Numerical experiments validate rapid convergence, best gradient-rate behavior in the LDG setting, and optimal primal-rate performance for $1<p\le2$, while demonstrating $hk$-independent iteration counts for the nonlinear solver across cases.

Abstract

We study the high-order local discontinuous Galerkin (LDG) method for the $p$-Laplace equation. We reformulate our spatial discretization as an equivalent convex minimization problem and use a preconditioned gradient descent method as the nonlinear solver. For the first time, a weighted preconditioner that provides $hk$-independent convergence is applied in the LDG setting. For polynomial order $k \geqslant 1$, we rigorously establish the solvability of our scheme and provide a priori error estimates in a mesh-dependent energy norm. Our error estimates are under a different and non-equivalent distance from existing LDG results. For arbitrarily high-order polynomials under the assumption that the exact solution has enough regularity, the error estimates demonstrate the potential for high-order accuracy. Our numerical results exhibit the desired convergence speed facilitated by the preconditioner, and we observe best convergence rates in gradient variables in alignment with linear LDG, and optimal rates in the primal variable when $1 < p \leqslant 2$.

A high-order local discontinuous Galerkin method for the $p$-Laplace equation

TL;DR

This work develops a high-order local discontinuous Galerkin method for the nonlinear -Laplace equation by reformulating the discretization as a convex minimization problem and solving it with a weighted preconditioned gradient descent to achieve -independent convergence. A discrete energy functional is minimized on a broken Sobolev space, with a DG gradient operator used to define the primal problem and gradient variables. The authors establish solvability, equivalence between the LDG weak form and the minimization form, and provide a priori error estimates in a mesh-dependent energy norm under sufficient regularity, revealing the potential for high-order accuracy. Numerical experiments validate rapid convergence, best gradient-rate behavior in the LDG setting, and optimal primal-rate performance for , while demonstrating -independent iteration counts for the nonlinear solver across cases.

Abstract

We study the high-order local discontinuous Galerkin (LDG) method for the -Laplace equation. We reformulate our spatial discretization as an equivalent convex minimization problem and use a preconditioned gradient descent method as the nonlinear solver. For the first time, a weighted preconditioner that provides -independent convergence is applied in the LDG setting. For polynomial order , we rigorously establish the solvability of our scheme and provide a priori error estimates in a mesh-dependent energy norm. Our error estimates are under a different and non-equivalent distance from existing LDG results. For arbitrarily high-order polynomials under the assumption that the exact solution has enough regularity, the error estimates demonstrate the potential for high-order accuracy. Our numerical results exhibit the desired convergence speed facilitated by the preconditioner, and we observe best convergence rates in gradient variables in alignment with linear LDG, and optimal rates in the primal variable when .
Paper Structure (26 sections, 27 theorems, 81 equations, 6 figures, 7 tables, 2 algorithms)

This paper contains 26 sections, 27 theorems, 81 equations, 6 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. The Governing Equation
  3. The p-Laplace Equation
  4. Variational Forms of the p-Laplace Equation
  5. Numerical Methods
  6. Derivation of the LDG Schemes
  7. Notations, Meshes, and Function Spaces
  8. Derivation of the Weak-Form LDG Scheme
  9. Conversion to the Minimization-Form LDG Scheme
  10. Solving the Minimization Problem
  11. Preconditioned Gradient Descent
  12. Line Search
  13. Theoretical Analysis
  14. Discrete Functional Analysis and FEM Tools
  15. Equivalence, Uniqueness and Existence
  16. ...and 11 more sections

Key Result

Proposition 2.1

When $p \in (1, +\infty)$, the primal weak form p-Laplace primal weak form and the minimization form p-Laplace primal minimization form are equivalent, and their solution exists and is unique.

Figures (6)

  • Figure 5.1: Uniformly refined conforming triangular meshes for a 2D polygonal domain.
  • Figure 5.2: Convergence history for Example \ref{['exm:re']}, with $\sigma = 0$ and $p = 1.5$.
  • Figure 5.3: Convergence history for Example \ref{['exm:re']}, with $\sigma = 7$ and $p = 4$.
  • Figure 5.4: Convergence history for Example \ref{['exm:de']}, with $p = 4$.
  • Figure 5.5: Convergence history for Example \ref{['exm:sm']}, with $p = 1.5$.
  • ...and 1 more figures

Theorems & Definitions (69)

  • Definition 2.1
  • Definition 2.2: Sobolev norms
  • Definition 2.3: Sobolev spaces
  • Definition 2.4: Conjugate index
  • Proposition 2.1: Existence, uniqueness and equivalence
  • proof
  • Lemma 2.1
  • Proposition 2.2: Energy estimates
  • proof
  • Definition 3.1: Broken Sobolev spaces
  • ...and 59 more