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Numerical solution — consistency, stability, convergence

Plan a numerical verification study using a reference problem, mesh refinement and error norms. A visually smooth solution is not evidence of convergence; stability and implementation checks address different risks.

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{
  "nodes": [
    { "id": "eq", "type": "rounded", "label": "Continuous model and conditions", "x": 40, "y": 200, "width": 150, "height": 104, "fill": "#ede9fe", "stroke": "#6d28d9" },
    { "id": "disc", "type": "rounded", "label": "Discretisation scheme", "x": 215, "y": 200, "width": 150, "height": 104, "fill": "#fef3c7", "stroke": "#b45309" },
    { "id": "impl", "type": "rounded", "label": "Implementation", "x": 390, "y": 200, "width": 150, "height": 104, "fill": "#fef3c7", "stroke": "#b45309" },
    { "id": "ref", "type": "octagon", "label": "Manufactured / reference solution", "x": 390, "y": 520, "width": 150, "height": 104, "fill": "#e0f2fe", "stroke": "#0369a1" },
    { "id": "mesh", "type": "rounded", "label": "Systematic mesh refinement", "x": 565, "y": 200, "width": 150, "height": 104, "fill": "#fef3c7", "stroke": "#b45309" },
    { "id": "err", "type": "card", "label": "Error norms and observed order", "x": 740, "y": 200, "width": 150, "height": 104, "fill": "#ccfbf1", "stroke": "#0f766e" },
    { "id": "stab", "type": "octagon", "label": "Stability / conservation checks", "x": 565, "y": 520, "width": 150, "height": 104, "fill": "#e0f2fe", "stroke": "#0369a1" },
    { "id": "conv", "type": "rounded", "label": "Convergence evidence in test scope", "x": 915, "y": 200, "width": 150, "height": 104, "fill": "#fce7f3", "stroke": "#be185d" }
  ],
  "edges": [
    { "id": "e0", "source": "eq", "target": "disc", "label": "approximate" },
    { "id": "e1", "source": "disc", "target": "impl", "label": "code" },
    { "id": "e2", "source": "impl", "target": "mesh", "label": "solve sequence" },
    { "id": "e3", "source": "ref", "target": "err", "label": "known comparison" },
    { "id": "e4", "source": "mesh", "target": "err", "label": "measure error" },
    { "id": "e5", "source": "stab", "target": "conv", "label": "boundedness checks" },
    { "id": "e6", "source": "err", "target": "conv", "label": "verify order" },
    { "id": "e7", "source": "eq", "target": "ref", "label": "construct test" }
  ]
}