Residual Tolerances and Discretization Error in Variable-Coefficient Diffusion: A Reproducible Study of Preconditioned Conjugate-Gradient Stopping Rules

This reproducible computational study examines the relationship between linear solver residuals, algebraic error and spatial discretization error in a conservative finite-difference approximation of two-dimensional diffusion. It includes a Python implementation of preconditioned conjugate gradients, identity/Jacobi/x-line preconditioners, two smooth coefficient profiles, two manufactured solution families, mesh-refinement studies, and 31 validation tests. The recorded experiment comprises 64 parameter cases, 192 solver trajectories, 960 fixed-tolerance observations and 1,536 total stopping-policy observations. Constant-coefficient cases are repeated controls under the two profile labels; policy observations are correlated within each trajectory. A fixed relative residual tolerance of 1e-4 exceeds the target algebraic-to-discretization error ratio of 0.1 in 24 of 192 recorded settings. A conservative residual-bound policy with a two-coarse-grid estimate meets this target throughout the tested suite, with a median 31.0% reduction in iterations relative to the 1e-10 baseline. Fixed 1e-6 also meets the target in every tested setting. Iteration reductions are not claims of end-to-end runtime speedup. The package contains the manuscript, source code, configuration, raw iteration traces, summary tables, figures, test evidence, environment metadata and reproduction scripts. It documents the limitations of the coarse-grid estimate and the unusually favorable Jacobi-scaled spectrum of the exponential profile. It does not claim a new conjugate-gradient algorithm or a universal estimator. Initial code, experiments and manuscript were prepared with OpenAI Codex assistance. Deepak Kumar's independent reproduction, substantive review and final contribution statement are pending in this author-review version. This work has not undergone peer review. Licensing: Python source code, tests and execution configuration are licensed under the MIT License. The manuscript, figures, numerical data and non-code documentation are licensed under Creative Commons Attribution 4.0 International (CC BY 4.0). These licenses apply to different file groups, not as alternative licenses for every file; see LICENSES.txt in the research archive for exact scope and attribution.