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A least squares finite element method for backward parabolic problems

Harald Monsuur

Abstract

Backward parabolic equations, such as the backward heat equation, are classical examples of ill-posed problems where solutions may not exist or depend continuously on the data. In this work, we study a least squares finite element method to numerically approximate solutions to such problems. We derive conditional stability estimates for the weak formulation of inhomogeneous backward parabolic equations, assuming minimal regularity of the solution. These stability results are then used to establish \emph{a priori} error bounds for our proposed method. We address key computational aspects, including the treatment of dual norms through the construction of suitable test spaces, and iterative solutions. Numerical experiments are used to validate our theoretical findings.

A least squares finite element method for backward parabolic problems

Abstract

Backward parabolic equations, such as the backward heat equation, are classical examples of ill-posed problems where solutions may not exist or depend continuously on the data. In this work, we study a least squares finite element method to numerically approximate solutions to such problems. We derive conditional stability estimates for the weak formulation of inhomogeneous backward parabolic equations, assuming minimal regularity of the solution. These stability results are then used to establish \emph{a priori} error bounds for our proposed method. We address key computational aspects, including the treatment of dual norms through the construction of suitable test spaces, and iterative solutions. Numerical experiments are used to validate our theoretical findings.
Paper Structure (35 sections, 10 theorems, 88 equations, 5 figures)

This paper contains 35 sections, 10 theorems, 88 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Conditional stability estimates
  3. Numerical solution
  4. Outline
  5. Notations
  6. Abstract framework and assumptions
  7. Abstract framework
  8. Additional assumptions
  9. Conditional stability
  10. Lipschitz continuity
  11. Conditional stability estimates for backward parabolic problems
  12. Density results
  13. Regularity estimate for $t>0$
  14. Case of Sobolev spaces
  15. Conditional stability estimates
  16. ...and 20 more sections

Key Result

Lemma 3.1

Let $u\in \mathcal{X}$. Let $\varphi_\varepsilon(t):= t-2\varepsilon \frac{t}{T}+\varepsilon$ and let $\rho\in C^\infty(\mathbb R)$ be a smooth function such that $\int_\mathbb R \rho=1$ and $\mathop{\mathrm{supp}}\nolimits \rho = (-1,1)$. Define then for $\varepsilon\to 0$ it holds that

Figures (5)

  • Figure 1: The case $d=2$. Left: $L_2(\Omega)$-error for different values of $t$ versus number of DoFs in $\mathcal{X}^\delta$. Right: $L_2(J\times \Omega)$ and $L_2(J;H^1(\Omega))$-errors versus number of DoFs in $\mathcal{X}^\delta$.
  • Figure 2: The case $d=3$. Left: $L_2(\Omega)$-error for different values of $t$ versus number of DoFs in $\mathcal{X}^\delta$. Right: $L_2(J\times \Omega)$ and $L_2(J;H^1(\Omega))$-errors versus number of DoFs in $\mathcal{X}^\delta$.
  • Figure 3: Error of numerical approximation at $t =\frac{15}{16}$ versus number of DoFs in $\mathcal{X}^\delta$ for different intervals $J$. Exact solution is $u(t,x,y) = \exp(2\pi^2(1-t))\sin(\pi x)\sin(\pi y)$.
  • Figure 4: Error of numerical approximation at $t =\frac{1}{16}$ versus number of DoFs in $\mathcal{X}^\delta$, in case of random perturbation of $g$.
  • Figure 5: Error of numerical approximation at $t =\frac{1}{16}$ versus number of DoFs in $\mathcal{X}^\delta$, in case of perturbation of $g$ with $0.05\cdot 2 \sin(\pi x)(\sin \pi y)$.

Theorems & Definitions (25)

  • remark 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3: Regularity estimate for $t>0$
  • proof
  • Definition 3.4
  • Lemma 3.5
  • proof
  • ...and 15 more