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Adaptive finite element methods for partial differential equations

Rolf Rannacher

TL;DR

The paper develops a goal-oriented, dual-weighted residual framework for a posteriori error estimation in Galerkin finite element methods, enabling efficient adaptivity for nonlinear variational problems including optimal control and eigenvalue computations. By embedding problems in a Lagrangian setting and using primal and dual residuals, it derives localized error representations with cubic remainders that drive mesh refinement toward the quantity of interest $J(u)$. The approach is applied to fluid flow simulations, demonstrating drag computation, drag minimization via boundary control, and stability analysis of optimized flows, with practical procedures that include solving linear dual problems and error balancing. This yields substantial computational savings and targeted accuracy in CFD and related PDE problems, illustrating the practical impact of the Dual Weighted Residual Method on producing accurate, resource-efficient simulations.

Abstract

The numerical simulation of complex physical processes requires the use of economical discrete models. This lecture presents a general paradigm of deriving a posteriori error estimates for the Galerkin finite element approximation of nonlinear problems. Employing duality techniques as used in optimal control theory the error in the target quantities is estimated in terms of weighted `primal' and `dual' residuals. On the basis of the resulting local error indicators economical meshes can be constructed which are tailored to the particular goal of the computation. The performance of this {\it Dual Weighted Residual Method} is illustrated for a model situation in computational fluid mechanics: the computation of the drag of a body in a viscous flow, the drag minimization by boundary control and the investigation of the optimal solution's stability.

Adaptive finite element methods for partial differential equations

TL;DR

The paper develops a goal-oriented, dual-weighted residual framework for a posteriori error estimation in Galerkin finite element methods, enabling efficient adaptivity for nonlinear variational problems including optimal control and eigenvalue computations. By embedding problems in a Lagrangian setting and using primal and dual residuals, it derives localized error representations with cubic remainders that drive mesh refinement toward the quantity of interest . The approach is applied to fluid flow simulations, demonstrating drag computation, drag minimization via boundary control, and stability analysis of optimized flows, with practical procedures that include solving linear dual problems and error balancing. This yields substantial computational savings and targeted accuracy in CFD and related PDE problems, illustrating the practical impact of the Dual Weighted Residual Method on producing accurate, resource-efficient simulations.

Abstract

The numerical simulation of complex physical processes requires the use of economical discrete models. This lecture presents a general paradigm of deriving a posteriori error estimates for the Galerkin finite element approximation of nonlinear problems. Employing duality techniques as used in optimal control theory the error in the target quantities is estimated in terms of weighted `primal' and `dual' residuals. On the basis of the resulting local error indicators economical meshes can be constructed which are tailored to the particular goal of the computation. The performance of this {\it Dual Weighted Residual Method} is illustrated for a model situation in computational fluid mechanics: the computation of the drag of a body in a viscous flow, the drag minimization by boundary control and the investigation of the optimal solution's stability.

Paper Structure

This paper contains 9 sections, 4 theorems, 38 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Paradigm of a posteriori error analysis
  3. Variational equations
  4. Optimal control problems
  5. Eigenvalue problems
  6. Application in fluid flow simulation
  7. Drag computation (from Becker00)
  8. Drag minimization (from Becker01)
  9. Stability of optimized flows (from HeuvelineRannacher02)

Key Result

Proposition 1

There holds the a posteriori error representation for arbitrary $\,y_h\in X_h$. The remainder $\,R_h\,$ is cubic in $\,e := x\!-\!x_h$,

Figures (5)

  • Figure 1: Refined meshes by 'ad hoc' strategy (top) and DWR method (bottom)
  • Figure 2: Velocity of the uncontrolled flow (top), controlled flow (middle), corresponding adapted mesh (bottom)
  • Figure 3: Streamlines of real parts of the 'critical' eigenfunction shortly before the Hopf bifurcation and after, depending on the imposed pressure drop
  • Figure 4: Real and imaginary parts of the critical eigenvalue as function of the control variable
  • Figure :

Theorems & Definitions (4)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4