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Applications of a space-time FOSLS formulation for parabolic PDEs

Gregor Gantner, Rob Stevenson

TL;DR

The paper demonstrates that a space-time first-order system least-squares (FOSLS) formulation for parabolic PDEs yields a symmetric, coercive bilinear form with a computable a posteriori error estimator, enabling efficient and certified solutions for parameter-dependent problems, optimal control, and time-dependent domains. By exploiting affine parameter separability, a reduced-basis method provides rapid online solutions with provable error control, while a saddle-point reformulation secures uniform inf-sup stability for a broad class of controls. The approach is extended to moving-domain problems via a Piola-type transformation on a fixed space-time mesh, avoiding ALE mappings. Numerical experiments in 1+1D and 2+1D spaces illustrate optimal convergence and robust error estimation, confirming the framework’s practicality for complex, parameterized, and dynamic-domain parabolic problems.

Abstract

In this work, we show that the space-time first-order system least-squares (FOSLS) formulation [Führer, Karkulik, Comput. Math. Appl. 92 (2021)] for the heat equation and its recent generalization [Gantner, Stevenson, ESAIM Math. Model. Numer. Anal. 55 (2021)] to arbitrary second-order parabolic PDEs can be used to efficiently solve parameter-dependent problems, optimal control problems, and problems on time-dependent spatial domains.

Applications of a space-time FOSLS formulation for parabolic PDEs

TL;DR

The paper demonstrates that a space-time first-order system least-squares (FOSLS) formulation for parabolic PDEs yields a symmetric, coercive bilinear form with a computable a posteriori error estimator, enabling efficient and certified solutions for parameter-dependent problems, optimal control, and time-dependent domains. By exploiting affine parameter separability, a reduced-basis method provides rapid online solutions with provable error control, while a saddle-point reformulation secures uniform inf-sup stability for a broad class of controls. The approach is extended to moving-domain problems via a Piola-type transformation on a fixed space-time mesh, avoiding ALE mappings. Numerical experiments in 1+1D and 2+1D spaces illustrate optimal convergence and robust error estimation, confirming the framework’s practicality for complex, parameterized, and dynamic-domain parabolic problems.

Abstract

In this work, we show that the space-time first-order system least-squares (FOSLS) formulation [Führer, Karkulik, Comput. Math. Appl. 92 (2021)] for the heat equation and its recent generalization [Gantner, Stevenson, ESAIM Math. Model. Numer. Anal. 55 (2021)] to arbitrary second-order parabolic PDEs can be used to efficiently solve parameter-dependent problems, optimal control problems, and problems on time-dependent spatial domains.
Paper Structure (27 sections, 5 theorems, 65 equations, 7 figures)

This paper contains 27 sections, 5 theorems, 65 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Parameter-dependent problems
  3. Optimal control problems
  4. Time-dependent domains
  5. Outline
  6. Preliminaries
  7. General notation
  8. Formulation of parabolic PDEs as first-order system
  9. Parameter-dependent problems
  10. Reduced basis method
  11. Basis generation
  12. Numerical experiment
  13. Optimal control problems
  14. Formulation as saddle-point problem
  15. Optimality system of PDEs
  16. ...and 12 more sections

Key Result

Theorem 2.1

With $X:= L_2(I;H_0^1(\Omega)) \cap H^1(I;H^{-1}(\Omega))$, $Y:= L_2(I;H_0^1(\Omega))$, and $\gamma_0(u):= u(0,\cdot)$ for all $u\in X,v\in Y$, the mapping is a linear isomorphism. In particular, there exists a unique $u \in X$ such that

Figures (7)

  • Figure 3.1: Approximation error of greedy algorithm for reduced basis method of Section \ref{['sec:reduced 1d']}.
  • Figure 3.2: Approximation errors at ${\boldsymbol \mu} = (\mu_1,0,0)$ with $\mu_1\in[0.5,1.5]$ (red) and ${\boldsymbol \mu} = (0.5,\mu_2,0.75)$ with $\mu_2\in[0,1]$ (blue) for reduced basis method of Section \ref{['sec:reduced 1d']} with $N=21$.
  • Figure 4.1: Optimal control problem in 1+1D of Section \ref{['sec:control 1d']}.
  • Figure 4.2: Optimal control problem in 2+1D of Section \ref{['sec:control 2d']}.
  • Figure 5.1: Space-time domains $Q$ of Section \ref{['sec:move 1d']} (left) and Section \ref{['sec:move 2d']} (right).
  • ...and 2 more figures

Theorems & Definitions (11)

  • Theorem 2.1
  • Theorem 2.2: gs21
  • Lemma 2.3: gs21
  • Remark 2.4
  • Remark 4.1
  • Lemma 4.2
  • proof
  • Remark 4.3
  • Theorem 5.1
  • proof
  • ...and 1 more