A Discretization-Accurate Stopping Criterion for Iterative Solvers for Finite Element Approximation

TL;DR

This paper proposes a new stopping criterion for iterative solvers in finite element methods that balances discretization and algebraic errors, reducing computational costs.

Contribution

It introduces a discretization-accurate stopping criterion based on duality and iterate differences, improving efficiency over traditional residual-based methods.

Findings

Significant reduction in computational cost.

Effective balancing of discretization and algebraic errors.

Validated with multigrid and Gauss-Seidel methods on Poisson problems.

Abstract

This paper introduces a discretization-accurate stopping criterion of symmetric iterative methods for solving systems of algebraic equations resulting from the finite element approximation. The stopping criterion consists of the evaluations of the discretization and the algebraic error estimators, that are based on the respective duality error estimator and the difference of two consecutive iterates. Iterations are terminated when the algebraic estimator is of the same magnitude as the discretization estimator. Numerical results for multigrid $V(1,1)$-cycle and symmetric Gauss-Seidel iterative methods are presented for the linear finite element approximation to the Poisson equations. A large reduction in computational cost is observed compared to the standard residual-based stopping criterion.