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Improved a priori error estimates for a space-time finite element method for parabolic problems

Thi Thanh Mai Ta, Quang Huy Nguyen, Phi Hung Pham

TL;DR

This paper addresses a parabolic initial-boundary value problem discretized with a space-time finite element method on fully unstructured space-time meshes. The authors develop a Petrov-Galerkin formulation and apply duality arguments (Aubin–Nitsche) to derive refined a priori error estimates in multiple norms, including $L^2(\Omega)$ at $t=T$, $L^2(\Omega_T)$, and the $H^1(\Omega_T)$-norm, with an additional negative-order norm result. They also prove discrete stability and projection properties, and validate the theory with numerical experiments that include moving-domain scenarios. The results provide sharper convergence guarantees for space-time FEM and suggest directions for extending the framework to other space-time methods and potential superconvergence phenomena.

Abstract

In this paper, we employ a space-time finite element method to discretize the parabolic initial-boundary value problem and extend its error analysis with refined estimates on unstructured space-time meshes. We establish higher-order estimates in three different norms, thereby supplementing existing research. Moreover, we obtain an optimal estimate in a norm stronger than that of the trial space. Finally, we present numerical examples to illustrate our theoretical results.

Improved a priori error estimates for a space-time finite element method for parabolic problems

TL;DR

This paper addresses a parabolic initial-boundary value problem discretized with a space-time finite element method on fully unstructured space-time meshes. The authors develop a Petrov-Galerkin formulation and apply duality arguments (Aubin–Nitsche) to derive refined a priori error estimates in multiple norms, including at , , and the -norm, with an additional negative-order norm result. They also prove discrete stability and projection properties, and validate the theory with numerical experiments that include moving-domain scenarios. The results provide sharper convergence guarantees for space-time FEM and suggest directions for extending the framework to other space-time methods and potential superconvergence phenomena.

Abstract

In this paper, we employ a space-time finite element method to discretize the parabolic initial-boundary value problem and extend its error analysis with refined estimates on unstructured space-time meshes. We establish higher-order estimates in three different norms, thereby supplementing existing research. Moreover, we obtain an optimal estimate in a norm stronger than that of the trial space. Finally, we present numerical examples to illustrate our theoretical results.

Paper Structure

This paper contains 5 sections, 8 theorems, 48 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Variational formulation and its discretization
  3. A priori error estimates
  4. Numerical results
  5. Conclusions

Key Result

Lemma 1

For all $y\in \mathop{\mathrm{H}}\nolimits^1\left(\Omega_T\right)$, the operator $\Pi_h$ satisfies the inequality Moreover, for all $\eta\in \left[1, k+1\right]$ and $y\in \mathop{\mathrm{H}}\nolimits^\eta\left(\Omega_T\right)$, the following estimate holds Here, $\mathop{\mathrm{D}}\nolimits:= \left(\nabla, \partial_t\right)^\top$ denotes the space-time gradient operator.

Figures (6)

  • Figure 1: The discrete solution $u_h$ for Example \ref{['ex: example 1']} with mesh size $h= 2^{-8}$.
  • Figure 2: Estimated convergence orders with respect to three different norms in Example \ref{['ex: example 1']}.
  • Figure 3: Estimated convergence orders with respect to three different norms in Example \ref{['ex: example 2']}.
  • Figure 4: Estimated convergence orders with respect to three different norms in Example \ref{['ex: example 3']}.
  • Figure 5: The discrete solution $u_h$ for Example \ref{['ex: example 4']} with mesh size $h= 2^{-8}$.
  • ...and 1 more figures

Theorems & Definitions (16)

  • Lemma 1
  • Lemma 2
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Corollary 1
  • Lemma 3
  • ...and 6 more