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Aubin--Nitsche-type estimates for space-time FOSLS for parabolic PDEs

Thomas Führer, Gregor Gantner

TL;DR

The paper develops Aubin--Nitsche-type (dual-based) $L^2(Q)$ error estimates for space-time first-order system least-squares (FOSLS) discretizations of the heat equation, showing that the $L^2$ error in the scalar variable can converge faster than the energy-like $U$-norm error under convexity and smooth data. It builds a commuting-diagram framework with specialized interpolation operators to obtain a bound of the form $\|u-u_h\|_{L^2(Q)} \lesssim \gamma(h)\|\boldsymbol{u}-\boldsymbol{u}_h\|_U$, plus a higher-order conservation bound for $\operatorname{div}_{t,\boldsymbol{x}}\boldsymbol{u}_h$ and a quasi-optimality result in a weaker norm via a projected data operator $\mathcal{Q}$. A detailed proof structure includes dual and primal-dual problems, and a careful use of interpolation-projection properties to achieve the bounds, with numerical experiments in 1D and 2D validating the predicted rates for smooth and non-smooth data. The results enhance understanding of how space-time FOSLS for parabolic PDEs can deliver higher accuracy in $L^2$-based quantities and inform practical choices for mesh-time refinement. Overall, the work strengthens the theoretical foundation and provides computational guidance for high-accuracy parabolic FOSLS methods.

Abstract

We develop Aubin--Nitsche-type estimates for recently proposed first-order system least-squares finite element methods (FOSLS) for the heat equation. Under certain assumptions, which are satisfied if the spatial domain is convex and the heat source and initial datum are sufficiently smooth, we prove that the $L^2$ error of approximations of the scalar field variable converges at a higher rate than the overall error. Furthermore, a higher-order conservation property is shown. In addition, we discuss quasi-optimality in weaker norms. Numerical experiments confirm our theoretical findings.

Aubin--Nitsche-type estimates for space-time FOSLS for parabolic PDEs

TL;DR

The paper develops Aubin--Nitsche-type (dual-based) error estimates for space-time first-order system least-squares (FOSLS) discretizations of the heat equation, showing that the error in the scalar variable can converge faster than the energy-like -norm error under convexity and smooth data. It builds a commuting-diagram framework with specialized interpolation operators to obtain a bound of the form , plus a higher-order conservation bound for and a quasi-optimality result in a weaker norm via a projected data operator . A detailed proof structure includes dual and primal-dual problems, and a careful use of interpolation-projection properties to achieve the bounds, with numerical experiments in 1D and 2D validating the predicted rates for smooth and non-smooth data. The results enhance understanding of how space-time FOSLS for parabolic PDEs can deliver higher accuracy in -based quantities and inform practical choices for mesh-time refinement. Overall, the work strengthens the theoretical foundation and provides computational guidance for high-accuracy parabolic FOSLS methods.

Abstract

We develop Aubin--Nitsche-type estimates for recently proposed first-order system least-squares finite element methods (FOSLS) for the heat equation. Under certain assumptions, which are satisfied if the spatial domain is convex and the heat source and initial datum are sufficiently smooth, we prove that the error of approximations of the scalar field variable converges at a higher rate than the overall error. Furthermore, a higher-order conservation property is shown. In addition, we discuss quasi-optimality in weaker norms. Numerical experiments confirm our theoretical findings.

Paper Structure

This paper contains 17 sections, 14 theorems, 108 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. General notation
  4. Preliminaries
  5. Bochner--Sobolev spaces
  6. Parabolic regularity
  7. Variational formulation of FOSLS
  8. Meshes and discretization space
  9. Interpolation operators
  10. Improved $L^2(Q)$ error estimates
  11. $L^\infty(I;L^2(\Omega))$ error estimate
  12. Notes on quasi-optimality in weaker norms
  13. Numerical experiments
  14. Smooth solution on unit interval
  15. Non-smooth solution on unit interval
  16. ...and 2 more sections

Key Result

Theorem 1

For given $f \in L^2(I;H^{-1}(\Omega))$ and $u_0\in L^2(\Omega)$, there exists a unique weak solution $u\in X$ of the heat equation eq:heat. It satisfies the stability estimate ∎

Figures (8)

  • Figure 1: Convergence plot for smooth solution on unit square of Section \ref{['sec:1d_sincos']} for parabolic meshes with $h_t\eqsim h_{\boldsymbol{x}}^2$.
  • Figure 2: Convergence plot for smooth solution on unit square of Section \ref{['sec:1d_sincos']} for parabolic meshes with $h_t\eqsim h_{\boldsymbol{x}}^2$.
  • Figure 3: Convergence plot for smooth solution on unit square of Section \ref{['sec:1d_f0_u0hat']} for parabolic meshes with $h_t\eqsim h_{\boldsymbol{x}}^2$.
  • Figure 4: Convergence plot for smooth solution on unit square of Section \ref{['sec:1d_f0_u0hat']} for parabolic meshes with $h_t\eqsim h_{\boldsymbol{x}}^2$.
  • Figure 5: Convergence plot for smooth solution on unit square of Section \ref{['sec:2d_square_sincos']} for parabolic meshes with $h_t\eqsim h_{\boldsymbol{x}}^2$.
  • ...and 3 more figures

Theorems & Definitions (29)

  • Theorem 1
  • Theorem 2
  • Proposition 3: Parabolic regularity
  • Remark 4
  • Remark 5
  • Lemma 6
  • proof
  • Proposition 7
  • Lemma 8
  • Lemma 9
  • ...and 19 more