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Finite element error analysis for elliptic parameter identification with power-type nonlinearity

De-Han Chen, Yi-Hsuan Lin, Irwin Yousept

Abstract

This paper studies the numerical analysis of a parameter identification problem governed by elliptic equations with power-type nonlinearity. We propose a numerical reconstruction via a suitable least-squares minimization problem based on piecewise linear finite elements. As one of our main novelties, we establish conditional stability estimates at the continuous level, which form the theoretical foundation of the present finite element analysis. Our stability analysis relies on tailored analytical tools, including Hardy-type inequalities, fractional Gagliardo-Nirenberg inequalities, and weighted spaces with singular distance weights. By invoking the achieved conditional stability together with the Carstensen quasi-interpolation operator and associated estimates in negative Sobolev spaces, we derive a priori error estimates for the proposed finite element approximation in terms of the mesh size, the regularization parameter, the noise level, and the nonlinearity exponent. Our results extend the recent stability and error estimates for the linear case by Jin et al. \cite{jin2022convergence} and sharpen their error estimates and convergence order under weaker regularity assumptions.

Finite element error analysis for elliptic parameter identification with power-type nonlinearity

Abstract

This paper studies the numerical analysis of a parameter identification problem governed by elliptic equations with power-type nonlinearity. We propose a numerical reconstruction via a suitable least-squares minimization problem based on piecewise linear finite elements. As one of our main novelties, we establish conditional stability estimates at the continuous level, which form the theoretical foundation of the present finite element analysis. Our stability analysis relies on tailored analytical tools, including Hardy-type inequalities, fractional Gagliardo-Nirenberg inequalities, and weighted spaces with singular distance weights. By invoking the achieved conditional stability together with the Carstensen quasi-interpolation operator and associated estimates in negative Sobolev spaces, we derive a priori error estimates for the proposed finite element approximation in terms of the mesh size, the regularization parameter, the noise level, and the nonlinearity exponent. Our results extend the recent stability and error estimates for the linear case by Jin et al. \cite{jin2022convergence} and sharpen their error estimates and convergence order under weaker regularity assumptions.
Paper Structure (10 sections, 13 theorems, 135 equations, 2 figures, 2 tables)

This paper contains 10 sections, 13 theorems, 135 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Well-posedness
  4. Stability analysis
  5. Analytical tools
  6. Conditional stability estimates
  7. Verification of Assumption \ref{['Assump: cse']}
  8. Finite element approximations
  9. Error estimates
  10. Numerical simulation and discussions

Key Result

Lemma 2.2

Let Assumption Assump: regularity be satisfied. Then, there exists a real number $\hat{p}>2$ such that for every $p \in (\hat{p}',\hat{p})$ the linear operator is a topological isomorphism.

Figures (2)

  • Figure 5.1: Exact and recovered coefficients for Example \ref{['numer:example']} (a)
  • Figure 5.2: Exact and recovered coefficients for Example \ref{['numer:example']} (b)

Theorems & Definitions (33)

  • Lemma 2.2: Meyer63_Lp and JK95_Dirichlet
  • Lemma 2.3: Well-posedness
  • proof
  • Lemma 2.4: Maximum principle and comparison principle
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Definition 3.3
  • ...and 23 more