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A posteriori error estimator for elliptic interface problems in the fictitious formulation

Najwa Alshehri, Daniele Boffi, Lucia Gastaldi

TL;DR

Explores a posteriori error estimation for elliptic interface problems solved with a fictitious-domain approach and a distributed Lagrange multiplier, using discontinuous multiplier spaces. Develops a residual-based estimator and proves global reliability and local efficiency for both constant and piecewise-smooth diffusion coefficients, including oscillation data terms. Validates the theory with immersed geometry simulations and strong coefficient jumps, demonstrating optimal adaptive convergence and robustness to geometric singularities. Provides practical error-control tools for unfitted FD-DLM methods and supports efficient mesh refinement in complex interface problems.

Abstract

A posteriori error estimator is derived for an elliptic interface problem in the fictitious domain formulation with distributed Lagrange multiplier considering a discontinuous Lagrange multiplier finite element space. A posteriori error estimation plays a pivotal role in assessing the accuracy and reliability of computational solutions across various domains of science and engineering. This study delves into the theoretical underpinnings and computational considerations of a residual-based estimator. Theoretically, the estimator is studied for cases with constant coefficients which jump across an interface as well as generalized scenarios with smooth coefficients that jump across an interface. Theoretical findings demonstrate the reliability and efficiency of the proposed estimators under all considered cases. Numerical experiments are conducted to validate the theoretical results, incorporating various immersed geometries and instances of high coefficients jumps at the interface. Leveraging an adaptive algorithm, the estimator identifies regions with singularities and applies refinement accordingly. Results substantiate the theoretical findings, highlighting the reliability and efficiency of the estimators. Furthermore, numerical solutions exhibit optimal convergence properties, demonstrating resilience against geometric singularities or coefficients jumps.

A posteriori error estimator for elliptic interface problems in the fictitious formulation

TL;DR

Explores a posteriori error estimation for elliptic interface problems solved with a fictitious-domain approach and a distributed Lagrange multiplier, using discontinuous multiplier spaces. Develops a residual-based estimator and proves global reliability and local efficiency for both constant and piecewise-smooth diffusion coefficients, including oscillation data terms. Validates the theory with immersed geometry simulations and strong coefficient jumps, demonstrating optimal adaptive convergence and robustness to geometric singularities. Provides practical error-control tools for unfitted FD-DLM methods and supports efficient mesh refinement in complex interface problems.

Abstract

A posteriori error estimator is derived for an elliptic interface problem in the fictitious domain formulation with distributed Lagrange multiplier considering a discontinuous Lagrange multiplier finite element space. A posteriori error estimation plays a pivotal role in assessing the accuracy and reliability of computational solutions across various domains of science and engineering. This study delves into the theoretical underpinnings and computational considerations of a residual-based estimator. Theoretically, the estimator is studied for cases with constant coefficients which jump across an interface as well as generalized scenarios with smooth coefficients that jump across an interface. Theoretical findings demonstrate the reliability and efficiency of the proposed estimators under all considered cases. Numerical experiments are conducted to validate the theoretical results, incorporating various immersed geometries and instances of high coefficients jumps at the interface. Leveraging an adaptive algorithm, the estimator identifies regions with singularities and applies refinement accordingly. Results substantiate the theoretical findings, highlighting the reliability and efficiency of the estimators. Furthermore, numerical solutions exhibit optimal convergence properties, demonstrating resilience against geometric singularities or coefficients jumps.
Paper Structure (12 sections, 5 theorems, 90 equations, 34 figures, 1 algorithm)

This paper contains 12 sections, 5 theorems, 90 equations, 34 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Notation
  4. Model Problem
  5. Continuous FD-DLM problem
  6. Discrete FD-DLM problem
  7. The a posteriori error estimator
  8. Constant diffusion coefficients
  9. Reliability and efficiency of the error estimator
  10. Piecewise smooth diffusion coefficients
  11. Adaptivity
  12. Numerical results

Key Result

Lemma 3.1

Let $\mathit{I}: H^1(\overline{\omega}_K) \rightarrow X_h$ be a Clément interpolation operator. Then, for any element $K$ in the partition $\mathcal{T}$, and any edge $E$ in the set of edges $\mathcal{E}$, there exists a constant $C>0$ that depends only the shape parameter of the partition, such tha The collection of these norms over all elements and edges leads to the following bounds

Figures (34)

  • Figure 1: Immersed boundary domains.
  • Figure 2: Fictitiously extended domain with immersed interface.
  • Figure 5: Extension of $\lambda_h$ out of $\Omega_2$.
  • Figure 6: $\overline{\omega}_{K_i}$.
  • Figure 7: $\omega_{K_i}$.
  • ...and 29 more figures

Theorems & Definitions (13)

  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Lemma 3.1
  • Proposition 3.1
  • proof
  • Definition 3.5
  • Lemma 3.2
  • Lemma 3.3
  • ...and 3 more