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Optimal Time-Adaptivity for Parabolic Problems with applications to Model Order Reduction

Michael Feischl, Fernando Henríquez, David Niederkofler

TL;DR

The paper addresses the lack of provable rate-optimal adaptivity for time-dependent parabolic PDEs by combining a uniformly inf-sup stable time stepping (Radau IIA hybrid scheme) with an adaptive time refinement framework. It proves that the adaptive time stepping is rate-optimal and extends the analysis to a model order reduction setting using Laplace-transform snapshots to build a tailored reduced space with exponential convergence. The resulting method yields a highly efficient space-time approximation that preserves accuracy with reduced complexity, and numerical experiments demonstrate both optimal time adaptivity and the effectiveness of MOR via Laplace snapshots. The work provides concrete error estimators, convergence theory, and practical MOR strategies, with implications for scalable simulation of parabolic systems.

Abstract

Since the first optimality proofs for adaptive mesh refinement algorithms in the early 2000s, the theory of optimal mesh refinement for PDEs was inherently limited to stationary problems. The reason for this is that time-dependent problems usually do not exhibit the necessary coercive structure that is used in optimality proofs to show a certain quasi-orthogonality, which is crucial for the theory. Recently, by using a new equivalence between quasi-orthogonality and inf-sup stability of the underlying problem, it was shown that an adaptive Crank-Nicolson scheme for the heat equation is optimal under a severe step size restriction. In this work, we use this new approach towards quasi-orthogonality together with a Radau IIA method that combines the advantages of the Crank-Nicolson and implicit Euler schemes. We obtain the first adaptive time stepping method for non-stationary PDEs that is provably rate optimal with respect to number of time steps vs. approximation error. Together with a reduced basis method that leverages the Laplace transform for building tailored subspaces of reduced dimension, we obtain a very efficient method.

Optimal Time-Adaptivity for Parabolic Problems with applications to Model Order Reduction

TL;DR

The paper addresses the lack of provable rate-optimal adaptivity for time-dependent parabolic PDEs by combining a uniformly inf-sup stable time stepping (Radau IIA hybrid scheme) with an adaptive time refinement framework. It proves that the adaptive time stepping is rate-optimal and extends the analysis to a model order reduction setting using Laplace-transform snapshots to build a tailored reduced space with exponential convergence. The resulting method yields a highly efficient space-time approximation that preserves accuracy with reduced complexity, and numerical experiments demonstrate both optimal time adaptivity and the effectiveness of MOR via Laplace snapshots. The work provides concrete error estimators, convergence theory, and practical MOR strategies, with implications for scalable simulation of parabolic systems.

Abstract

Since the first optimality proofs for adaptive mesh refinement algorithms in the early 2000s, the theory of optimal mesh refinement for PDEs was inherently limited to stationary problems. The reason for this is that time-dependent problems usually do not exhibit the necessary coercive structure that is used in optimality proofs to show a certain quasi-orthogonality, which is crucial for the theory. Recently, by using a new equivalence between quasi-orthogonality and inf-sup stability of the underlying problem, it was shown that an adaptive Crank-Nicolson scheme for the heat equation is optimal under a severe step size restriction. In this work, we use this new approach towards quasi-orthogonality together with a Radau IIA method that combines the advantages of the Crank-Nicolson and implicit Euler schemes. We obtain the first adaptive time stepping method for non-stationary PDEs that is provably rate optimal with respect to number of time steps vs. approximation error. Together with a reduced basis method that leverages the Laplace transform for building tailored subspaces of reduced dimension, we obtain a very efficient method.

Paper Structure

This paper contains 24 sections, 26 theorems, 182 equations, 4 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Model Problem & Discretization
  3. Radau IIA: A hybrid Euler/Crank-Nicolson scheme
  4. Error estimator in time
  5. The adaptive algorithm
  6. Optimality of the adaptive time stepping
  7. Equivalent Petrov-Galerkin formulation
  8. Inf-sup stability
  9. Proof of the main result
  10. Application to Model Order Reduction for Parabolic Problems
  11. MOR of Parabolic Problem using the Laplace Transform
  12. Exponential Convergence
  13. Fully Discrete Error Estimate
  14. Cost Comparison
  15. Numerical Experiments
  16. ...and 9 more sections

Key Result

Lemma 1

Given $u\in{\mathcal{X}}$, the test function $v:=A^{-1}(\partial_t u + Au)\in {\mathcal{Y}}$ satisfies and $\|v\|_{{\mathcal{Y}}}\leq C\|u\|_{{\mathcal{X}}}$, with constants $C,c_0,c_1>0$ that are independent of $u$ and ${t_{\rm end}}$.

Figures (4)

  • Figure 1: Figure \ref{['fig:Test_1_a_Error']}. Convergence of the error in the $\mathcal{X}$-norm (comparison with finest approximation) and estimator for adaptive ($\theta = 1/2$) and uniform mesh refinement. Figure \ref{['fig:Test_1_a_Error_Mesh_Size']}. Sizes of local time steps of the last iteration of the adaptive/uniform algorithm plotted over their position in the time interval $[0, 1]$.
  • Figure 2: Figure \ref{['fig:Test_1_b_g_1']}. Convergence of the error in the $\mathcal{X}_h$-norm (comparison with the finest approximation) and estimator for adaptive ($\theta = 1/2$) and uniform mesh refinement with the mesh closure procedure and $g_0 \in \{0.9,0.99\}$. Figure \ref{['fig:Test_SVD']}. Singular values of the snapshot matrix for $M \in \{50,75,100,125\}$.
  • Figure 3: Comparison of the Crank-Nicolson (CN) and the proposed hybrid Euler/Crank-Nicolson scheme (Hybrid) for adaptive ($\theta = 1/2$) mesh refinement and for different values of the total number of degrees of freedom $N_h$. The convergence of the error is computed in the $\mathcal{X}_h$-norm (comparison with the finest approximation). The meshes $\mathcal{E}_h$ are obtained through a successive uniform refinement of a given starting mesh.
  • Figure 4: Convergence of the model order reduction techniques based on the Laplace transform for $R \in \{5,10,15,20\}$. The initial condition corresponds to the $H^1(\Omega)$-projection of $u_0$ onto $\mathcal{S}^1_0(\mathcal{E}_h).$

Theorems & Definitions (49)

  • Lemma 1
  • Theorem 1
  • Lemma 2
  • Proof 1
  • Theorem 3
  • Lemma 4
  • Proof 2
  • Corollary 5
  • Lemma 6
  • Lemma 7
  • ...and 39 more