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Optimal finite element approximation of unique continuation

Erik Burman, Mihai Nechita, Lauri Oksanen

TL;DR

A recently introduced class of finite element methods with weakly consistent regularisation is recalled and the associated error estimates are shown to be quasi optimal in the sense of the definition.

Abstract

We consider finite element approximations of ill-posed elliptic problems with conditional stability. The notion of {\emph{optimal error estimates}} is defined including both convergence with respect to mesh parameter and perturbations in data. The rate of convergence is determined by the conditional stability of the underlying continuous problem and the polynomial order of the finite element approximation space. A proof is given that no finite element approximation can converge at a better rate than that given by the definition, justifying the concept. A recently introduced class of finite element methods with weakly consistent regularisation is recalled and the associated error estimates are shown to be quasi optimal in the sense of our definition.

Optimal finite element approximation of unique continuation

TL;DR

A recently introduced class of finite element methods with weakly consistent regularisation is recalled and the associated error estimates are shown to be quasi optimal in the sense of the definition.

Abstract

We consider finite element approximations of ill-posed elliptic problems with conditional stability. The notion of {\emph{optimal error estimates}} is defined including both convergence with respect to mesh parameter and perturbations in data. The rate of convergence is determined by the conditional stability of the underlying continuous problem and the polynomial order of the finite element approximation space. A proof is given that no finite element approximation can converge at a better rate than that given by the definition, justifying the concept. A recently introduced class of finite element methods with weakly consistent regularisation is recalled and the associated error estimates are shown to be quasi optimal in the sense of our definition.
Paper Structure (9 sections, 6 theorems, 92 equations)

This paper contains 9 sections, 6 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Unique continuation problem
  3. Motivation and outline
  4. Optimal error estimates for elliptic problems
  5. Optimal three-ball estimate
  6. Definition of optimal convergence for ill-posed problems with conditional stability
  7. Proof of optimality
  8. Primal-dual finite element methods with weakly consistent regularisation
  9. Conclusion

Key Result

theorem 1

Let $n\in\{2,3\}$ and $B(r) \subset \mathbb R^n$ be the open ball of radius $r > 0$. Let $0 < r_1 < r_2 < r_3$. Then for all harmonic functions $u$ there holds where Moreover, there does not exist $\tilde{\alpha} > \alpha$ such that

Theorems & Definitions (19)

  • theorem 1
  • proof
  • remark thmcounterremark
  • definition thmcounterdefinition
  • remark thmcounterremark
  • remark thmcounterremark
  • lemma thmcounterlemma
  • proof
  • theorem 2
  • proof
  • ...and 9 more