Stopping Criteria for the Conjugate Gradient Algorithm in High-Order Finite Element Methods

TL;DR

This paper proposes new stopping criteria for the conjugate gradient algorithm in high-order finite element methods, balancing algebraic and discretization errors, with proven reliability and demonstrated efficiency through numerical experiments.

Contribution

Introduces a residual-based stopping criterion that scales with mesh size and polynomial degree, and a subdomain-based criterion for variable coefficients, with theoretical guarantees and practical validation.

Findings

The new criteria prevent premature termination and over-solving.

The residual-based criterion is robust across mesh sizes and polynomial degrees.

Numerical tests confirm efficiency in complex, variable-coefficient problems.

Abstract

We consider stopping criteria that balance algebraic and discretization errors for the conjugate gradient algorithm applied to high-order finite element discretizations of Poisson problems. Firstly, we introduce a new stopping criterion that suggests stopping when the norm of the linear system residual is less than a small fraction of an error indicator derived directly from the residual. This indicator shares the same mesh size and polynomial degree scaling as the norm of the residual, resulting in a robust criterion regardless of the mesh size, the polynomial degree, and the shape regularity of the mesh. Secondly, for solving Poisson problems with highly variable piecewise constant coefficients, we introduce a subdomain-based criterion that recommends stopping when the norm of the linear system residual restricted to each subdomain is smaller than the corresponding indicator also restricted to that subdomain. Reliability and efficiency theorems for the first criterion are established. Numerical experiments, including tests with highly variable piecewise constant coefficients and a GPU-accelerated three-dimensional elliptic solver, demonstrate that the proposed criteria efficiently avoid both premature termination and over-solving.