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Adaptive Mesh Refinement for Variational Inequalities

Giuliano Stefano Fochesatto

TL;DR

This work tackles the challenge of solving variational inequalities efficiently by concentrating mesh refinement near the free boundary. It introduces two AMR schemes, Variable Coefficient Elliptic Smoothing (VCES) and Unstructured Dilation Operator (UDO), implemented in the Firedrake framework, and analyzes how accurate free-boundary localization affects VI solvers. The methods leverage node-wise indicators, a time-stepping smoothing step, and graph-based neighbor dilation to produce refined meshes without hanging nodes, improving convergence rates and solver performance on obstacle-type problems. Empirical results on canonical obstacle problems (including ball and spiral obstacles) show that VCES and UDO achieve near-best refinement quality (as measured by active-set metrics) with significantly fewer degrees of freedom, and that the approaches offer parallel-load balancing benefits for large-scale computations. The work lays groundwork for hybrid and metric-based refinements in more complex free-boundary VI contexts.

Abstract

Variational inequalities play a pivotal role in a wide array of scientific and engineering applications. This project presents two techniques for adaptive mesh refinement (AMR) in the context of variational inequalities, with a specific focus on the classical obstacle problem. We propose two distinct AMR strategies: Variable Coefficient Elliptic Smoothing (VCES) and Unstructured Dilation Operator (UDO). VCES uses a nodal active set indicator function as the initial iterate to a time-dependent heat equation problem. Solving a single step of this problem has the effect of smoothing the indicator about the free boundary. We threshold this smoothed indicator function to identify elements near the free boundary. Key parameters such as timestep and threshold values significantly influence the efficacy of this method. The second strategy, UDO, focuses on the discrete identification of elements adjacent to the free boundary, employing a graph-based approach to mark neighboring elements for refinement. This technique resembles the dilation morphological operation in image processing, but tailored for unstructured meshes. We also examine the theory of variational inequalities, the convergence behavior of finite element solutions, and implementation in the Firedrake finite element library. Convergence analysis reveals that accurate free boundary estimation is pivotal for solver performance. Numerical experiments demonstrate the effectiveness of the proposed methods in dynamically enhancing mesh resolution around free boundaries, thereby improving the convergence rates and computational efficiency of variational inequality solvers. Our approach integrates seamlessly with existing Firedrake numerical solvers, and it is promising for solving more complex free boundary problems.

Adaptive Mesh Refinement for Variational Inequalities

TL;DR

This work tackles the challenge of solving variational inequalities efficiently by concentrating mesh refinement near the free boundary. It introduces two AMR schemes, Variable Coefficient Elliptic Smoothing (VCES) and Unstructured Dilation Operator (UDO), implemented in the Firedrake framework, and analyzes how accurate free-boundary localization affects VI solvers. The methods leverage node-wise indicators, a time-stepping smoothing step, and graph-based neighbor dilation to produce refined meshes without hanging nodes, improving convergence rates and solver performance on obstacle-type problems. Empirical results on canonical obstacle problems (including ball and spiral obstacles) show that VCES and UDO achieve near-best refinement quality (as measured by active-set metrics) with significantly fewer degrees of freedom, and that the approaches offer parallel-load balancing benefits for large-scale computations. The work lays groundwork for hybrid and metric-based refinements in more complex free-boundary VI contexts.

Abstract

Variational inequalities play a pivotal role in a wide array of scientific and engineering applications. This project presents two techniques for adaptive mesh refinement (AMR) in the context of variational inequalities, with a specific focus on the classical obstacle problem. We propose two distinct AMR strategies: Variable Coefficient Elliptic Smoothing (VCES) and Unstructured Dilation Operator (UDO). VCES uses a nodal active set indicator function as the initial iterate to a time-dependent heat equation problem. Solving a single step of this problem has the effect of smoothing the indicator about the free boundary. We threshold this smoothed indicator function to identify elements near the free boundary. Key parameters such as timestep and threshold values significantly influence the efficacy of this method. The second strategy, UDO, focuses on the discrete identification of elements adjacent to the free boundary, employing a graph-based approach to mark neighboring elements for refinement. This technique resembles the dilation morphological operation in image processing, but tailored for unstructured meshes. We also examine the theory of variational inequalities, the convergence behavior of finite element solutions, and implementation in the Firedrake finite element library. Convergence analysis reveals that accurate free boundary estimation is pivotal for solver performance. Numerical experiments demonstrate the effectiveness of the proposed methods in dynamically enhancing mesh resolution around free boundaries, thereby improving the convergence rates and computational efficiency of variational inequality solvers. Our approach integrates seamlessly with existing Firedrake numerical solvers, and it is promising for solving more complex free boundary problems.

Paper Structure

This paper contains 28 sections, 6 theorems, 89 equations, 26 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. A Brief Introduction to the Finite Element Method
  3. Reference Problem: Poisson Equation on Square Domain
  4. Weak Formulation
  5. Galerkin Finite Element Method
  6. Firedrake Implementation: Poisson Equation on Square Domain
  7. Theory of Errors
  8. Introduction to Variational Inequalities
  9. Reference Problem: Ball Obstacle Problem
  10. Numerical Methods for Variational Inequalities
  11. Active-Set Newton's Method
  12. Firedrake Implementation: Ball Obstacle
  13. Convergence of Variational Inequalities
  14. Adaptive Mesh Refinement Techniques for PDE Problems
  15. Tag and Refine Methods
  16. ...and 13 more sections

Key Result

Theorem 1

Given a linear variational problem (2.22), with solution $u$, and Galerkin approximation $u_h$ solving (2.23), then for all $v_h \in S^h_0$ we have that,

Figures (26)

  • Figure 1: Plot of Reference Problem, surface plot (left), contour plot (right).
  • Figure 2: Basis functions on Dirichlet boundary node (left) and interior node (right).
  • Figure 3: Convergence of reference problem
  • Figure 4: Refinement schemes.
  • Figure 5: Radial cross section of $u(r)$.
  • ...and 21 more figures

Theorems & Definitions (11)

  • Definition 1
  • Theorem 1
  • proof
  • Theorem 2
  • Theorem 3
  • proof
  • Theorem 4
  • Definition 2
  • Theorem 5
  • Theorem 6
  • ...and 1 more