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Locally different models in a checkerboard pattern with mesh adaptation and error control for multiple quantities of interest

Bernhard Endtmayer

TL;DR

This work develops a multi-goal, goal-oriented a posteriori error estimation framework using the dual weighted residual (DWR) method for finite element PDEs with locally varying models arranged in a checkerboard. By combining partition-of-unity localization with a single aggregated QoI $J_c$ for multiple quantities of interest, the authors establish an adaptive algorithm that concentrates mesh refinement where the combined error is largest and near QoI hotspots. Numerical results demonstrate robust error control (effectivity between $0.4$ and $1.6$) and favorable convergence rates under adaptive refinement compared to uniform meshes, validating the approach for complex, heterogeneous PDE systems. The methodology offers a practical route to efficiently solve PDEs with multiple QoIs across domains featuring distinct local physics.

Abstract

In this work, we apply multi-goal oriented error estimation to the finite element method. In particular, we use the dual weighted residual method and apply it to a model problem. This model problem consist of locally different coercive partial differential equations in a checkerboard pattern, where the solution is continuous across the interface. In addition to the error estimation, the error can be localized using a partition of unity technique. The resulting adaptive algorithm is substantiated with a numerical example.

Locally different models in a checkerboard pattern with mesh adaptation and error control for multiple quantities of interest

TL;DR

This work develops a multi-goal, goal-oriented a posteriori error estimation framework using the dual weighted residual (DWR) method for finite element PDEs with locally varying models arranged in a checkerboard. By combining partition-of-unity localization with a single aggregated QoI for multiple quantities of interest, the authors establish an adaptive algorithm that concentrates mesh refinement where the combined error is largest and near QoI hotspots. Numerical results demonstrate robust error control (effectivity between and ) and favorable convergence rates under adaptive refinement compared to uniform meshes, validating the approach for complex, heterogeneous PDE systems. The methodology offers a practical route to efficiently solve PDEs with multiple QoIs across domains featuring distinct local physics.

Abstract

In this work, we apply multi-goal oriented error estimation to the finite element method. In particular, we use the dual weighted residual method and apply it to a model problem. This model problem consist of locally different coercive partial differential equations in a checkerboard pattern, where the solution is continuous across the interface. In addition to the error estimation, the error can be localized using a partition of unity technique. The resulting adaptive algorithm is substantiated with a numerical example.
Paper Structure (7 sections, 1 theorem, 13 equations, 9 figures, 1 table)

This paper contains 7 sections, 1 theorem, 13 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Model Problem
  3. Discretization
  4. Goal Oriented Adaptivity
  5. Multi Goal Oriented Adaptivity
  6. Numerical Results
  7. Conclusions

Key Result

Theorem 4.1

If $J(u) \in \mathbb{R}$, where $u\in V$ solves our model problem eq: cont primal problem, $u_h^{(2)} \in V_h^{(2)}$ solves the enriched problem eq: enriched primal problem and $z_h^{(2)} \in V_h^{(2)}$ solves the enriched adjoint problem eq: enriched adjoint problem, then for arbitrary but fixed $\ holds, where $\rho(\tilde{u})(\cdot) := -\mathcal{A}(\tilde{u})(\cdot)$ , $\rho^*(\tilde{u},\tilde{

Figures (9)

  • Figure 1: The locally different models of the partial differential equations on the domain $\Omega$ (left) and the sub domains $\Omega_i$ for $i \in \{1,2,3,4\}$ with the interface $I$ (right) .
  • Figure 2: The solution $u$ (left) and the adaptive mesh after 31 adaptive refinement steps (right).
  • Figure 3: The quantities of interest in the table (left) and the error distribution resulting from $\eta_h$ with partition of unity localization on an 8 times uniformly refined grid. .
  • Figure 4: The localized primal part $\frac{1}{2}\rho(u_h)(z_h^{(2)})$ (left) and adjoint part $\frac{1}{2}\rho^*(u_h,z_h)(u_h^{(2)}-u_h)$ (right) of the error estimator $\eta_h$.
  • Figure 5: Effectivity index for $J_{c}$.
  • ...and 4 more figures

Theorems & Definitions (3)

  • Theorem 4.1: see EndtLaRiSchafWi24_book_chapterEnLaSchaf2023
  • proof
  • Remark 4.2