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Numerical modeling and open-source implementation of variational partition-of-unity localizations of space-time dual-weighted residual estimators for parabolic problems

Jan Philipp Thiele, Thomas Wick

TL;DR

The paper develops partition-of-unity dual-weighted residual estimators for space-time parabolic problems to enable goal-oriented adaptive refinement in both space and time. It extends PU localization from stationary to space-time settings and provides joint and split error estimators, including equal low/high order discretization strategies and various PU choices. The approach is demonstrated on linear heat equations and a nonlinear combustion model, with open-source implementations that demonstrate accurate error estimation and effective adaptivity. The results show the method's accuracy, robustness, and practical potential for multiphysics applications requiring efficient space-time error control.

Abstract

In this work, we consider space-time goal-oriented a posteriori error estimation for parabolic problems. Temporal and spatial discretizations are based on Galerkin finite elements of continuous and discontinuous type. The main objectives are the development and analysis of space-time estimators, in which the localization is based on a weak form employing a partition-of-unity. The resulting error indicators are used for temporal and spatial adaptivity. Our developments are substantiated with several numerical examples.

Numerical modeling and open-source implementation of variational partition-of-unity localizations of space-time dual-weighted residual estimators for parabolic problems

TL;DR

The paper develops partition-of-unity dual-weighted residual estimators for space-time parabolic problems to enable goal-oriented adaptive refinement in both space and time. It extends PU localization from stationary to space-time settings and provides joint and split error estimators, including equal low/high order discretization strategies and various PU choices. The approach is demonstrated on linear heat equations and a nonlinear combustion model, with open-source implementations that demonstrate accurate error estimation and effective adaptivity. The results show the method's accuracy, robustness, and practical potential for multiphysics applications requiring efficient space-time error control.

Abstract

In this work, we consider space-time goal-oriented a posteriori error estimation for parabolic problems. Temporal and spatial discretizations are based on Galerkin finite elements of continuous and discontinuous type. The main objectives are the development and analysis of space-time estimators, in which the localization is based on a weak form employing a partition-of-unity. The resulting error indicators are used for temporal and spatial adaptivity. Our developments are substantiated with several numerical examples.
Paper Structure (44 sections, 16 theorems, 111 equations, 12 figures, 14 tables, 2 algorithms)

This paper contains 44 sections, 16 theorems, 111 equations, 12 figures, 14 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Space-time notation and problem formulations
  3. Notation and spaces
  4. Discretization
  5. Semi-discretization in time
  6. Fully discrete abstract problem
  7. Heat equation
  8. Combustion
  9. General formulation of parabolic problems
  10. Numerical solution
  11. The dual weighted residual method in a space-time setting
  12. Error representation and estimation
  13. DWR for nonlinear time dependent problems
  14. Adjoint problem statements
  15. Goal-oriented error representations
  16. ...and 29 more sections

Key Result

Theorem 3.2

Let the primal problem and adjoint problem be given. Let $(u,z)\in \widetilde{X}(\mathcal{T}_k,V)\times \widetilde{X}(\mathcal{T}_k,V)$, $(u_k,z_k)\in \widetilde{X}_k^r(\mathcal{T}_k,V) \times \widetilde{X}_k^r(\mathcal{T}_k,V)$ and $(u_{kh},z_{kh})\in \widetilde{X}_{k,h}^{r,s}(\mathcal{T}_k,\mathca with the primal error estimator $\rho$ and the adjoint error estimator $\rho^*$ as well as a remain

Figures (12)

  • Figure 1: Different interpolation levels on a single 1+1D space-time element
  • Figure 2: Section \ref{['sec:config_hartmann_heat']}: Error convergence of the Hartmann testcase
  • Figure 3: Section \ref{['sec:config_hartmann_heat']}: Grid after 4 refinement loops with the split PU-DWR estimator at $t=i/4$, $i\in\{1,2,3,4\}$
  • Figure 4: Section \ref{['sec:config_hartmann_heat']}: Solution after 4 refinement loops with the split PU-DWR estimator at $t=i/4$, $i\in\{1,2,3,4\}$
  • Figure 5: Section \ref{['sec:config_hartmann_heat']}: Error convergence of the Hartmann testcase for $\eta^{1/1}$ and $M_\text{init} = 1600$
  • ...and 7 more figures

Theorems & Definitions (42)

  • Remark 2.4
  • Remark 2.5
  • Remark 3.1
  • Theorem 3.2: Joint error identity
  • proof
  • Theorem 3.3: Split error identity
  • proof
  • Proposition 4.1
  • proof
  • Remark 4.2
  • ...and 32 more