Review of Output-Based Error Estimation and Mesh Adaptation in Computational Fluid Dynamics

TLDR

Error estimation and control, particularly via adjoint‑based techniques that provide output‑specific error estimates and local refinement indicators, are essential for reliable CFD simulations, yet challenges such as computable error bounds and robust mesh adaptation persist. This article reviews recent adjoint‑based output‑error estimation and mesh adaptation techniques applied to CFD in aerospace engineering. The review derives the adjoint as the sensitivity of an output to residual perturbations, presents fully discrete and variational adjoint equations and the adjoint‑weighted residual method, discusses underlying assumptions, compares finite‑volume and finite‑element formulations, and surveys mesh‑adaptation strategies based on output‑error indicators. Recent adaptive studies on laminar and RANS flows demonstrate that output‑based adaptive methods substantially enhance CFD robustness.

Abstract

Error estimation and control are critical ingredients for improving the reliability of computational simulations. Adjoint-based techniques can be used to both estimate the error in chosen solution outputs and to provide local indicators for adaptive refinement. This article reviews recent work on these techniques for computational fluid dynamics applications in aerospace engineering. The definition of the adjoint as the sensitivity of an output to residual source perturbations is used to derive both the adjoint equation, in fully discrete and variational formulations, and the adjoint-weighted residual method for error estimation. Assumptions and approximations made in the calculations are discussed. Presentation of the discrete and variational formulations enables a side-by-side comparison of recent work in output-error estimation using the finite volume method and the finite element method. Techniques for adapting meshes using output-error indicators are also reviewed. Recent adaptive results from a variety of laminar and Reynolds-averaged Navier-Stokes applications show the power of output-based adaptive methods for improving the robustness of computational fluid dynamics computations. However, challenges and areas of additional future research remain, including computable error bounds and robust mesh adaptation mechanics.