We introduce three new stopping criteria that balance algebraic and discretization errors for the conjugate gradient algorithm applied to high-order finite element discretizations of Poisson problems. The current state of the art stopping criteria compare a posteriori estimates of discretization error against estimates of the algebraic error. Firstly, we propose a new error indicator derived from a recovery-based error estimator that is less computationally expensive and more reliable. Secondly, we introduce a new stopping criterion that suggests stopping when the norm of the linear 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. Thirdly, in solving Poisson problems with highly variable piecewise constant coefficients, we introduce a subdomain-based criterion that recommends stopping when the norm of the linear residual restricted to each subdomain is smaller than the corresponding indicator also restricted to that subdomain. Numerical experiments, including tests with anisotropic meshes and highly variable piecewise constant coefficients, demonstrate that the proposed criteria efficiently avoid both premature termination and over-solving.

翻译：针对泊松问题的高阶有限元离散，我们提出了三种新的共轭梯度算法停止准则，用于平衡代数误差与离散化误差。当前最先进的停止准则通过比较离散化误差的后验估计值与代数误差估计值进行判定。首先，我们基于恢复型误差估计器提出了一种新的误差指示器，该指示器计算成本更低且更可靠。其次，我们引入了一种新的停止准则：当线性残差范数小于直接从残差导出的误差指示器的一个小分数时建议停止。该指示器与残差范数具有相同的网格尺寸和多项式次数缩放特性，因此无论网格尺寸、多项式次数及网格形状正则性如何，均能保持准则的鲁棒性。第三，在求解具有高度可变分片常数系数的泊松问题时，我们提出了一种基于子域的准则：当限制在每个子域上的线性残差范数小于同样限制在该子域的对应指示器时建议停止。数值实验（包括各向异性网格和高度可变分片常数系数的测试）表明，所提出的准则能有效避免过早终止和过度求解。