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Anisotropic hp space-time adaptivity and goal-oriented error control for convection-dominated problems

Nils Margenberg, Marius Paul Bruchhäuser, Bernhard Endtmayer

Abstract

We present an anisotropic goal-oriented error estimator based on the Dual Weighted Residual (DWR) method for time-dependent convection-dominated problems. Using elementwise p-anisotropic finite element spaces, the estimator is elementwise separated with respect to the single directions in space and time. This naturally leads to adaptive, anisotropic hp-refinement (h-anisotropic refinement and elementwise anisotropic p-enrichment). We employ discontinuous elements in space and time, which are well suited for problems with high Peclet 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 hp-refinements that efficiently capture sharp layers. Numerical examples in up to three spatial dimensions demonstrate the superior performance of the proposed method compared to isotropic h and hp adaptive refinement using established benchmarks for convection-dominated transport.

Anisotropic hp space-time adaptivity and goal-oriented error control for convection-dominated problems

Abstract

We present an anisotropic goal-oriented error estimator based on the Dual Weighted Residual (DWR) method for time-dependent convection-dominated problems. Using elementwise p-anisotropic finite element spaces, the estimator is elementwise separated with respect to the single directions in space and time. This naturally leads to adaptive, anisotropic hp-refinement (h-anisotropic refinement and elementwise anisotropic p-enrichment). We employ discontinuous elements in space and time, which are well suited for problems with high Peclet 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 hp-refinements that efficiently capture sharp layers. Numerical examples in up to three spatial dimensions demonstrate the superior performance of the proposed method compared to isotropic h and hp adaptive refinement using established benchmarks for convection-dominated transport.
Paper Structure (34 sections, 7 theorems, 107 equations, 10 figures, 6 tables, 1 algorithm)

This paper contains 34 sections, 7 theorems, 107 equations, 10 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Mathematical Problem and Notation
  3. Convection-Diffusion-Reaction Equation.
  4. Weak Formulation of the CDR Equation.
  5. Discrete Spaces
  6. Anisotropic Polynomial Degrees in Space
  7. Anisotropic Polynomial Degrees in Time
  8. Time Discretization Matrices
  9. Space-Time Finite Element Discretization
  10. Discrete Space-Time Variational Formulation.
  11. Penalty parameter scaling.
  12. Local Algebraic Problem.
  13. Global Algebraic Problem
  14. Adjoint Algebraic System
  15. A Posteriori Error Representation
  16. ...and 19 more sections

Key Result

Theorem 5.1

Let $(u, z)$, $(u_\tau, z_\tau)$, and $(u_{\tau h}, z_{\tau h})$ be stationary points of $\mathcal{L}$, $\mathcal{L}_\tau$, and $\mathcal{L}_{\tau h}$, respectively. Then, for arbitrary $(\tilde{u}_\tau, \tilde{z}_\tau)$ in the time-semi-discrete space and $(\tilde{u}_{\tau h}, \tilde{z}_{\tau h})$ where $\mathcal{R}_\tau$ and $\mathcal{R}_h$ are higher-order remainders with respect to $(u - u_\t

Figures (10)

  • Figure 1: Visualization of the sets ${N_{p_K+1}^{({k})}}$ for $d=2$. The union of all sets is ${N({p_K+1})}$.
  • Figure 2: Sketch of the linear solution process: The space-time system on $I_n$ is preconditioned by a GMRES method to which we apply a block preconditioner based on the diagonalization of $\operatorname{tril}({(\boldsymbol{M}_{\tau}^{k_n})}^{-1}\boldsymbol{A}_\tau)$. The diagonal blocks are then solved with few iterations of a preconditioned GMRES method.
  • Figure 3: Geometry and coarse, unstructured mesh of the domain (left) and the best adaptive solution obtained for the $L^2$ space-time error control. On the left, green coloring corresponds to inhomogeneous Dirichlet BCs.
  • Figure 4: Anisotropic $hp$-mesh after $18$ refinement steps (cf. Tables \ref{['tab:step-layer-point-value']}) with diffusion coefficient $\varepsilon=10^{-6}$. The green squares are zoomed in to visualize the high anisotropic $h$ and $p$ refinement around the goal point. Within the mesh we observe high anisotropies in the mesh size and polynomial degree.
  • Figure 5: Comparison between adaptive anisotropic refinement strategies: anisotropic $hp$ refinement considering all polynomials $p, k \in \{1,\dots,9\}$ and only even polynomials $p, k \in \{2,4,6,8,10\}$. To validate the efficiency of the anisotropic $hp$ refinement we consider $h$ anisotropic refinement with isotropic linear and quadratic polynomial degree (uniform in space and time).
  • ...and 5 more figures

Theorems & Definitions (27)

  • Definition 3.1: $p$-Anisotropic Finite Element
  • Definition 3.2: Global $p$-Anisotropic Finite Element Spaces
  • Definition 3.3: $p$-Anisotropic Discrete Space-Time Function Spaces.
  • Remark 3.4: Tensor product function spaces
  • Theorem 5.1: Goal-oriented error representation
  • Definition 6.1: The restriction
  • Definition 6.2: The directional restriction
  • Definition 6.3: Isotropic Error Part
  • Lemma 6.4
  • proof
  • ...and 17 more