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Goal Oriented Adaptive Space Time Finite Element Methods Applied to Touching Domains

Bernhard Endtmayer, Andreas Schafelner

TL;DR

The paper tackles accurate computation of QoIs for parabolic PDEs on moving, touching domains by developing a goal-oriented adaptive space-time finite element method driven by the dual-weighted residual (DWR) approach. Time is treated as a spatial-like variable in an all-at-once formulation, enabling simultaneous refinement in space and time and leveraging adjoint sensitivities to steer mesh refinement toward the QoI. A discretized adjoint problem and an enriched primal space are used to form a computable estimator $\eta_h$ that is proven efficient and reliable, with localization via partition of unity. Numerical experiments on a moving-domain model demonstrate substantive convergence gains: for linear QoIs, adaptive refinement with $k=1$ achieves $O(h^{5/3})$ and with $k=2$ improves to $O(h^{13/6})$, while refinements concentrate at singular regions induced by contact and separation as well as near the QoI region.

Abstract

We consider goal-oriented adaptive space-time finite-element discretizations of the parabolic heat equation on completely unstructured simplicial space-time meshes. In some applications, we are interested in an accurate computation of some possibly nonlinear functionals at the solution, so called goal functionals. This motivates the use of adaptive mesh refinements driven by the dual-weighted residual (DWR) method. The DWR method requires the numerical solution of a linear adjoint problem that provides the sensitivities for the mesh refinement. This can be done by means of the same full space-time finite element discretization as used for the primal linear problem. The numerical experiment presented demonstrates that this goal-oriented, full space-time finite element solver efficiently provides accurate numerical results for a model problem with moving domains and a linear goal functional, where we know the exact value.

Goal Oriented Adaptive Space Time Finite Element Methods Applied to Touching Domains

TL;DR

The paper tackles accurate computation of QoIs for parabolic PDEs on moving, touching domains by developing a goal-oriented adaptive space-time finite element method driven by the dual-weighted residual (DWR) approach. Time is treated as a spatial-like variable in an all-at-once formulation, enabling simultaneous refinement in space and time and leveraging adjoint sensitivities to steer mesh refinement toward the QoI. A discretized adjoint problem and an enriched primal space are used to form a computable estimator that is proven efficient and reliable, with localization via partition of unity. Numerical experiments on a moving-domain model demonstrate substantive convergence gains: for linear QoIs, adaptive refinement with achieves and with improves to , while refinements concentrate at singular regions induced by contact and separation as well as near the QoI region.

Abstract

We consider goal-oriented adaptive space-time finite-element discretizations of the parabolic heat equation on completely unstructured simplicial space-time meshes. In some applications, we are interested in an accurate computation of some possibly nonlinear functionals at the solution, so called goal functionals. This motivates the use of adaptive mesh refinements driven by the dual-weighted residual (DWR) method. The DWR method requires the numerical solution of a linear adjoint problem that provides the sensitivities for the mesh refinement. This can be done by means of the same full space-time finite element discretization as used for the primal linear problem. The numerical experiment presented demonstrates that this goal-oriented, full space-time finite element solver efficiently provides accurate numerical results for a model problem with moving domains and a linear goal functional, where we know the exact value.
Paper Structure (6 sections, 13 equations, 3 figures, 1 algorithm)

This paper contains 6 sections, 13 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. The model problem and discretization
  3. The model problem
  4. Discretization
  5. Goal oriented error estimates
  6. Numerical experiments

Figures (3)

  • Figure 1: Two dimensional space-time domain, with the spatial dimension on the $x$-axis, and the temporal dimension on the $y$-axis.
  • Figure 2: Convergence rates of the error in the functional and the corresponding efficiency index for the space-time domain with pre-contact part.
  • Figure 3: Initial space-time solution $u_h$ with mesh (upper left); initial space-time sensitivity $z_h$ with mesh (upper right); space-time solution $u_h$ with mesh after 5 adaptive refinements (lower left); space-time sensitivity $z_h$ with mesh after 5 adaptive refinements (lower right); using linear finite elements and $k$-enrichment for the DWR-estimator. The mesh was stretched in $t$-direction for visualization purposes only.