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Plain convergence of goal-oriented adaptive FEM

Valentin Helml, Michael Innerberger, Dirk Praetorius

TL;DR

This work establishes plain convergence for GOAFEM across two abstract settings, relaxing reliability/efficiency requirements in the second setting. It introduces a generalized marking framework that couples primal and dual estimators via a function V, encompassing and extending classical marking strategies. The main results show that, under a priori convergence of the discrete primal and dual solutions, the product estimator $\eta_\ell \zeta_\ell$ tends to zero, ensuring convergence of the goal error estimators for Poisson-type models, nonlinear goals, and semilinear problems. Theoretical findings are complemented by numerical experiments illustrating marking strategies and GOAFEM performance, with practical impact on reliable and efficient goal-oriented adaptivity.

Abstract

We discuss goal-oriented adaptivity in the frame of conforming finite element methods and plain convergence of the related a posteriori error estimator for different general marking strategies. We present an abstract analysis for two different settings. First, we consider problems where a local discrete efficiency estimate holds. Second, we show plain convergence in a setting that relies only on structural properties of the error estimators, namely stability on non-refined elements as well as reduction on refined elements. In particular, the second setting does not require reliability and efficiency estimates. Numerical experiments underline our theoretical findings.

Plain convergence of goal-oriented adaptive FEM

TL;DR

This work establishes plain convergence for GOAFEM across two abstract settings, relaxing reliability/efficiency requirements in the second setting. It introduces a generalized marking framework that couples primal and dual estimators via a function V, encompassing and extending classical marking strategies. The main results show that, under a priori convergence of the discrete primal and dual solutions, the product estimator tends to zero, ensuring convergence of the goal error estimators for Poisson-type models, nonlinear goals, and semilinear problems. Theoretical findings are complemented by numerical experiments illustrating marking strategies and GOAFEM performance, with practical impact on reliable and efficient goal-oriented adaptivity.

Abstract

We discuss goal-oriented adaptivity in the frame of conforming finite element methods and plain convergence of the related a posteriori error estimator for different general marking strategies. We present an abstract analysis for two different settings. First, we consider problems where a local discrete efficiency estimate holds. Second, we show plain convergence in a setting that relies only on structural properties of the error estimators, namely stability on non-refined elements as well as reduction on refined elements. In particular, the second setting does not require reliability and efficiency estimates. Numerical experiments underline our theoretical findings.
Paper Structure (32 sections, 26 theorems, 142 equations, 4 figures, 1 algorithm)

This paper contains 32 sections, 26 theorems, 142 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Goal-oriented adaptive FEM
  3. An abstract approach to plain convergence of GOAFEM
  4. Outline
  5. Notation
  6. Abstract setting
  7. Discretization and mesh-refinement
  8. Goal-oriented error estimation
  9. Marking strategy
  10. Estimator properties
  11. Setting I based on discrete local efficiency
  12. Setting II based on high-level structural properties
  13. Motivating example: Poisson equation in Rd̂
  14. Discretization
  15. Error estimators
  16. ...and 17 more sections

Key Result

Proposition 6

There exists a constant $C_{\mathrm{rel}} > 0$ such that, for all $\mathcal{T}_H \in \mathbb{T}$, there holds The constant $C_{\mathrm{rel}}$ depends only on $\kappa$-shape regularity.

Figures (4)

  • Figure 1: General scheme of assumptions made in this work. The assumptions about the PDE model as well as some of the assumptions on mesh-refinement (denoted by "R") are general. Setting I requires further assumptions about mesh-refinement and involved norms (denoted by "N"). Finally, the assumptions on the estimator in Setting I and II are denoted by "A" and "B", respectively.
  • Figure 2: Goal error estimator $\eta_\ell \, \zeta_\ell$ over cumulative computational cost for the problem from Section \ref{['subsec:ms-example']}. The colors correspond to the marking strategies (i)--(iv), the markers correspond to different polynomial degrees of the FEM ansatz space $\mathcal{X}_\ell$.
  • Figure 3: Goal error estimator $\eta_\ell \, \zeta_\ell$ over cumulative computational cost for the problem from Section \ref{['subsec:bip-example']} with different polynomial degrees of the FEM ansatz space $\mathcal{X}_\ell$. Left: Goal error bound with sum structure. Right: Sharper goal error bound with product structure.
  • Figure 4: Meshes after several steps of the adaptive algorithm driven by $\eta_\ell \, \zeta_\ell = \mu_\ell \, [\mu_\ell^2 + \nu_\ell^2]^{1/2}$ for the problem from Section \ref{['subsec:bip-example']}. The weight of the goal functional is centered at $(0.5, 0.5)$. Left: Computation with $p=1$, $\#\mathcal{T}_6 = 3093$. Right: Computation with $p=3$, $\#\mathcal{T}_{14} = 2821$.

Theorems & Definitions (44)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Proposition 6: Reliability
  • Proposition 7: Local discrete efficiency
  • Proposition 8: Stability
  • Proposition 9: Reduction
  • Remark 10
  • ...and 34 more