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A fully iterative adaptive energy-based approach for monotone elliptic problems

Raphael Leu, Thomas P. Wihler

Abstract

We present a fully iterative adaptive algorithm for the numerical minimization of strongly convex energy functionals in Hilbert spaces. The proposed approach, which we first present in abstract form, generates a hierarchical sequence of adaptively refined finite-dimensional approximation spaces and employs a (nonlinear) conjugate gradient (CG) method to compute suitable approximations on each space. A core novelty of our approach is that all components of the algorithm are consistently driven by energy reduction principles rather than by classical a posteriori estimators. In particular, adaptive refinement is steered by local energy reduction indicators which aim to construct subsequent approximation spaces in a way that attains the largest potential decrease in energy. Likewise, the stopping criteria for the iterative solver are based on either relative or averaged energy reductions on each subspace. As a concrete realization, we present a concise implementation for $\mathbb{P}_1$ finite element discretizations of second-order semilinear elliptic diffusion-reaction models, where the local indicators driving the element refinements are computed based on edge-wise energy reductions. Numerical experiments demonstrate that the resulting scheme achieves optimal convergence for various benchmark problems in two-dimensional polygonal domains.

A fully iterative adaptive energy-based approach for monotone elliptic problems

Abstract

We present a fully iterative adaptive algorithm for the numerical minimization of strongly convex energy functionals in Hilbert spaces. The proposed approach, which we first present in abstract form, generates a hierarchical sequence of adaptively refined finite-dimensional approximation spaces and employs a (nonlinear) conjugate gradient (CG) method to compute suitable approximations on each space. A core novelty of our approach is that all components of the algorithm are consistently driven by energy reduction principles rather than by classical a posteriori estimators. In particular, adaptive refinement is steered by local energy reduction indicators which aim to construct subsequent approximation spaces in a way that attains the largest potential decrease in energy. Likewise, the stopping criteria for the iterative solver are based on either relative or averaged energy reductions on each subspace. As a concrete realization, we present a concise implementation for finite element discretizations of second-order semilinear elliptic diffusion-reaction models, where the local indicators driving the element refinements are computed based on edge-wise energy reductions. Numerical experiments demonstrate that the resulting scheme achieves optimal convergence for various benchmark problems in two-dimensional polygonal domains.
Paper Structure (20 sections, 2 theorems, 64 equations, 10 figures, 3 algorithms)

This paper contains 20 sections, 2 theorems, 64 equations, 10 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Iterative Solver And Stopping Criteria
  3. A Simplified Model Problem
  4. Galerkin Discretization
  5. Iterative Solver
  6. Stopping Criteria
  7. Application to more general minimization problems
  8. Space enrichment and variational adaptivity
  9. Variational Adaptivity
  10. Edge-based Variational Adaptivity
  11. The $\mathbb{P}_1$ Finite Element Space
  12. Edge-Based Space Enrichments
  13. Computational Aspects
  14. Mesh Refinement
  15. Computation of energy decay
  16. ...and 5 more sections

Key Result

Lemma 1

Let $\mathbb{W} \subset \mathbb{V}$ be any linear subspace, and $u_{\mathbb{W}} \in \mathbb{W}$ the unique minimizer of $\mathsf{E}$ over $\mathbb{W}$. Then, for all $w \in \mathbb{W}$, the identity holds true. Further, setting $w = u_{\mathbb{W}}$ in eq:galerkin-orthogonality-on-general-subspace immediately yields i.e. the energy of $u_{\mathbb{W}}$ is necessarily non-positive.

Figures (10)

  • Figure 1: Local patches associated to an interior edge $E = (z_i, z_j)$. Left: local patch $\omega_E$ before the local refinement. Right: local modified patch $\widetilde{\omega}_E$ obtained by bisection of the edge $E$ and removing the hanging node $\bar{z}_{ij}$ by connecting it to both vertices $z_{\sharp}$ and $z_{\flat}$.
  • Figure 2: Convergence plot for Experiment \ref{['experiment-nonlinear-1']}, obtained with Algorithm \ref{['alg:full-solver']} and three different stopping criteria. The black dashed line represents the optimal convergence rate $\mathcal{O}(n_{\text{DOF}}^{-1})$.
  • Figure 3: Convergence plot for Experiment \ref{['experiment-nonlinear-2']}, obtained with Algorithm \ref{['alg:full-solver']} and three different stopping criteria. The black dashed line represents the optimal convergence rate $\mathcal{O}(n_{\text{DOF}}^{-1})$.
  • Figure 4: Convergence plot for Experiment \ref{['experiment-nonlinear-3']}, obtained with Algorithm \ref{['alg:full-solver']} and three different stopping criteria. The black dashed line represents the optimal convergence rate $\mathcal{O}(n_{\text{DOF}}^{-1})$.
  • Figure 5: Convergence plot for Experiment \ref{['experiment-nonlinear-4']}, obtained with Algorithm \ref{['alg:full-solver']} and three different stopping criteria. The black dashed line represents the optimal convergence rate $\mathcal{O}(n_{\text{DOF}}^{-1})$.
  • ...and 5 more figures

Theorems & Definitions (10)

  • Lemma 1
  • proof
  • Remark 1
  • Lemma 2
  • proof
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6