Automated goal-oriented error control I: stationary variational problems

TL;DR

This paper introduces an automated framework for goal-oriented error control in nonlinear stationary finite element problems, combining linearization, error estimation, and adaptive mesh refinement to efficiently meet accuracy goals.

Contribution

It presents a novel automated approach integrating linearization, error estimation, and adaptive refinement for goal-oriented error control in nonlinear PDEs.

Findings

Effective error control demonstrated on Poisson's equation

Successful application to linear elasticity and Navier-Stokes equations

Automated process improves computational efficiency and accuracy

Abstract

This article presents a general and novel approach to the automation of goal-oriented error control in the solution of nonlinear stationary finite element variational problems. The approach is based on automated linearization to obtain the linearized dual problem, automated derivation and evaluation of a posteriori error estimates, and automated adaptive mesh refinement to control the error in a given goal functional to within a given tolerance. Numerical examples representing a variety of different discretizations of linear and nonlinear partial differential equations are presented, including Poisson's equation, a mixed formulation of linear elasticity, and the incompressible Navier-Stokes equations.