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

TL;DR

This paper introduces an anisotropic hp adaptive finite element method with goal-oriented error control for convection-dominated problems, effectively capturing sharp layers and improving computational efficiency.

Contribution

It develops a novel anisotropic goal-oriented error estimator using the DWR method with elementwise anisotropic p-refinement and h-refinement, tailored for convection-dominated problems.

Findings

Efficiently captures sharp layers in convection-dominated problems.

Demonstrates superior performance over isotropic refinement methods.

Robustness shown across different goal functionals and dimensions.

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.