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Anisotropic space-time goal-oriented error control and mesh adaptivity for convection-diffusion-reaction equations

M. Bause, M. Bruchhäuser, B. Endtmayer, N. Margenberg, I. Toulopoulos, T. Wick

TL;DR

This work develops an anisotropic space-time goal-oriented error estimator for time-dependent convection-diffusion-reaction equations using the Dual Weighted Residual (DWR) framework. By incorporating anisotropic interpolation and directional error indicators, the method yields mesh refinement that concentrates along dominant solution features in space and time, while SUPG stabilization mitigates spurious oscillations at high Péclet numbers. The proposed framework decouples directional error contributions and extends traditional isotropic DWR by introducing directional estimators and anisotropic restriction/interpolation operators, leading to substantial efficiency gains. Numerical experiments on interior and boundary layer benchmarks demonstrate superior accuracy per degree of freedom compared with isotropic refinement and global mesh strategies, validating the approach's potential for complex convection-dominated problems and scalable space-time adaptivity.

Abstract

We present an anisotropic goal-oriented error estimator based on the Dual Weighted Residual (DWR) method for time-dependent convection-diffusion-reaction (CDR) equations. Using anisotropic interpolation operators the estimator is elementwise separated with respect to the single directions in space and time leading to adaptive, anisotropic mesh refinement in a natural way. To prevent spurious oscillations the streamline upwind Petrov-Galerkin (SUPG) method is applied to stabilize the underlying system in the case of high Péclet numbers. Efficiency and robustness of the underlying algorithm are demonstrated for different goal functionals. The directional error indicators quantify anisotropy of the solution with respect to the goal, and produce meshes that efficiently capture sharp layers. Numerical examples show the superiority of the proposed approach over isotropic adaptive and global mesh refinement using established benchmarks for convection-dominated transport.

Anisotropic space-time goal-oriented error control and mesh adaptivity for convection-diffusion-reaction equations

TL;DR

This work develops an anisotropic space-time goal-oriented error estimator for time-dependent convection-diffusion-reaction equations using the Dual Weighted Residual (DWR) framework. By incorporating anisotropic interpolation and directional error indicators, the method yields mesh refinement that concentrates along dominant solution features in space and time, while SUPG stabilization mitigates spurious oscillations at high Péclet numbers. The proposed framework decouples directional error contributions and extends traditional isotropic DWR by introducing directional estimators and anisotropic restriction/interpolation operators, leading to substantial efficiency gains. Numerical experiments on interior and boundary layer benchmarks demonstrate superior accuracy per degree of freedom compared with isotropic refinement and global mesh strategies, validating the approach's potential for complex convection-dominated problems and scalable space-time adaptivity.

Abstract

We present an anisotropic goal-oriented error estimator based on the Dual Weighted Residual (DWR) method for time-dependent convection-diffusion-reaction (CDR) equations. Using anisotropic interpolation operators the estimator is elementwise separated with respect to the single directions in space and time leading to adaptive, anisotropic mesh refinement in a natural way. To prevent spurious oscillations the streamline upwind Petrov-Galerkin (SUPG) method is applied to stabilize the underlying system in the case of high Péclet numbers. Efficiency and robustness of the underlying algorithm are demonstrated for different goal functionals. The directional error indicators quantify anisotropy of the solution with respect to the goal, and produce meshes that efficiently capture sharp layers. Numerical examples show the superiority of the proposed approach over isotropic adaptive and global mesh refinement using established benchmarks for convection-dominated transport.

Paper Structure

This paper contains 20 sections, 4 theorems, 64 equations, 16 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model Problem and Variational Formulation
  3. Space-Time Discretization and Isotropic Error Representation
  4. Discretization in Time
  5. Discretization in Space and Stabilization
  6. Isotropic Goal Oriented Error Estimation
  7. Isotropic Patch-Wise Higher Order Interpolation
  8. Local Error Indicators in Space and Time
  9. Temporal Weights.
  10. Spatial Weights.
  11. Anisotropic Method
  12. Anisotropic Finite Elements
  13. Anisotropic Interpolation and Restriction Operations
  14. Anisotropic Goal Oriented Error Estimation
  15. Algorithm for Goal-Oriented Anisotropic Mesh Adaptivity
  16. ...and 5 more sections

Key Result

Theorem 3.3

Let $\{u,\,z\}\in \mathcal{V} \times \mathcal{V}$, $\{u_{\tau},z_{\tau}\} \in \mathcal{V}_{\tau}^{r} \times \mathcal{V}_{\tau}^{r}$, and $\{u_{\tau h},z_{\tau h}\} \in \mathcal{V}_{\tau h}^{r,p} \times \mathcal{V}_{\tau h}^{r,p}$ denote the stationary points of $\mathcal{L}, \mathcal{L}_{\tau}$, and Then, for the discretization errors in space and time we get the representation formulas where $\r

Figures (16)

  • Figure 1: DoFs of $Q_{1}$ finite elements on the patch $K_{2h}^P$ (left) and a the DoFs of a $Q_{2}$ finite element on the element $K_{2h}$ (right).
  • Figure 2: The function $v_h$.
  • Figure 3: The resulting function $\mathrm{I}_{2h}^{\left(2p\right)}v_h$.
  • Figure 5: $Q_{2,1}$ finite elements on $K_{2h,1^\ast}^P$.
  • Figure 6: $Q_{1,2}$ finite elements on $K_{2h,2^\ast}^P$.
  • ...and 11 more figures

Theorems & Definitions (22)

  • Definition 3.1
  • Remark 3.2
  • Theorem 3.3
  • Definition 3.4
  • Definition 3.5
  • Definition 3.6
  • Definition 4.1
  • Remark 4.2
  • Definition 4.3
  • Definition 4.4
  • ...and 12 more