Goal-oriented error analysis of iterative Galerkin discretizations for nonlinear problems including linearization and algebraic errors

TL;DR

This paper develops goal-oriented error estimates for iterative solvers of nonlinear PDEs, ensuring adjoint consistency, and introduces an adaptive algorithm that balances discretization and solver errors, validated by numerical examples.

Contribution

It introduces a novel goal-oriented error analysis framework for linearized iterative solvers that maintains adjoint consistency in nonlinear PDE discretizations.

Findings

The adaptive algorithm effectively balances discretization and iterative errors.

Numerical examples demonstrate the efficiency of the proposed error control method.

The approach guarantees adjoint consistency for improved error estimation.

Abstract

We consider the goal-oriented error estimates for a linearized iterative solver for nonlinear partial differential equations. For the adjoint problem and iterative solver we consider, instead of the differentiation of the primal problem, a suitable linearization which guarantees the adjoint consistency of the numerical scheme. We derive error estimates and develop an efficient adaptive algorithm which balances the errors arising from the discretization and use of iterative solvers. Several numerical examples demonstrate the efficiency of this algorithm.