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Numerical Study of Approximation Techniques for the Temporal Weights to the DWR Method

Marius Paul Bruchhäuser, Markus Bause

TL;DR

The paper addresses goal-oriented a posteriori error control for a time-dependent convection-diffusion-reaction problem using the Dual Weighted Residual (DWR) method. It analyzes two temporal-weight approximation strategies, hoRe and hoFE, within a space-time adaptive framework employing a discontinuous Galerkin discretization in time, continuous Galerkin in space, and SUPG stabilization. The main contributions are the derivation of space-time error representations, localization into time-slabs, and a detailed numerical comparison showing effectivity indices near 1 for both approaches, with hoFE offering slightly better stability at higher cost. The practical impact lies in providing robust, efficient, and accurate adaptive schemes for convection-dominated transient problems, enabling reliable goal-oriented refinement in applications requiring precise time-dependent quantities of interest.

Abstract

This work presents a numerical investigation of different approximation techniques for the temporal weights used in the Dual Weighted Residual (DWR) method applied to a time-dependent convection-diffusion equation which is assumed to be convection-dominated. It is a continuation of a previous work by the authors where spatial weights were compared for a steady-state case. A higher-order finite elements approach is compared to a more cost-efficient higher-order reconstruction approach. Numerical examples point out the results regarding accuracy, efficiency and stability reasons.

Numerical Study of Approximation Techniques for the Temporal Weights to the DWR Method

TL;DR

The paper addresses goal-oriented a posteriori error control for a time-dependent convection-diffusion-reaction problem using the Dual Weighted Residual (DWR) method. It analyzes two temporal-weight approximation strategies, hoRe and hoFE, within a space-time adaptive framework employing a discontinuous Galerkin discretization in time, continuous Galerkin in space, and SUPG stabilization. The main contributions are the derivation of space-time error representations, localization into time-slabs, and a detailed numerical comparison showing effectivity indices near 1 for both approaches, with hoFE offering slightly better stability at higher cost. The practical impact lies in providing robust, efficient, and accurate adaptive schemes for convection-dominated transient problems, enabling reliable goal-oriented refinement in applications requiring precise time-dependent quantities of interest.

Abstract

This work presents a numerical investigation of different approximation techniques for the temporal weights used in the Dual Weighted Residual (DWR) method applied to a time-dependent convection-diffusion equation which is assumed to be convection-dominated. It is a continuation of a previous work by the authors where spatial weights were compared for a steady-state case. A higher-order finite elements approach is compared to a more cost-efficient higher-order reconstruction approach. Numerical examples point out the results regarding accuracy, efficiency and stability reasons.
Paper Structure (9 sections, 1 theorem, 17 equations, 3 figures, 2 tables)

This paper contains 9 sections, 1 theorem, 17 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Variational Formulation and Discretization in Space and Time
  3. DWR-Based A Posteriori Error Estimation
  4. Proof.
  5. Practical Aspects
  6. Numerical Examples
  7. Example 1: Rotating Hill with Changing Orientation
  8. Example 2: Moving Hump with Circular Layer
  9. Conclusion

Key Result

Theorem 3.1

Let $\{u,z\} \in V \times V$, $\{u_{\tau},z_{\tau}\} \in V_{\tau}^{r} \times V_{\tau}^{r}$, and $\{u_{\tau h},z_{\tau h}\} \in V_{\tau h}^{r,p} \times V_{\tau h}^{r,p}$ be stationary points of Lagrangian functionals $\mathcal{L}, \mathcal{L}_{\tau}$, and $\mathcal{L}_{\tau h}$ on different discretiz Then, there holds the error representation formulas in space and time, respectively, where $\rho_\

Figures (3)

  • Figure 4.1: Reconstruction of a discontinuous constant (left) and linear (right) in time function using Gauss quadrature points.
  • Figure 4.2: Restriction of a discontinuous linear (left) and quadratic (right) in time function using Gauss quadrature points.
  • Figure 5.1: Comparison of effectivity indices $\mathcal{I}_{\mathrm{eff}}$ (over total space-time DoFs) regarding both approximation approaches for varying diffusion coefficients $\varepsilon$, using $\delta_0=1$, $\omega = 2$ and $J_T$ for Example 2.

Theorems & Definitions (2)

  • Theorem 3.1: Temporal and Spatial Error Representation
  • Remark 6.1