{
  "problem": "Cubic-Hermite FEM for Euler-Bernoulli beam u''''=f, clamped BCs",
  "exact_solution": "exp(x)*x^2*(1-x)^2",
  "f": "exp(x)*(x^4+14x^3+49x^2+32x-12)",
  "dofs_per_node": [
    "value",
    "slope"
  ],
  "nodal_study": {
    "description": "L-inf error at the nodes; nodally superconvergent",
    "table": [
      {
        "n": 32,
        "h": 0.03125,
        "e_n": 4.4803605270260505e-12,
        "beta": null
      },
      {
        "n": 64,
        "h": 0.015625,
        "e_n": 5.975775430044905e-14,
        "beta": 6.2283450853657385
      },
      {
        "n": 128,
        "h": 0.0078125,
        "e_n": 1.0145898010627263e-11,
        "beta": -7.407554915356821
      },
      {
        "n": 256,
        "h": 0.00390625,
        "e_n": 1.1037171177008531e-11,
        "beta": -0.12147389398932085
      },
      {
        "n": 512,
        "h": 0.001953125,
        "e_n": 5.380799417142867e-10,
        "beta": -5.60737816531502
      },
      {
        "n": 1024,
        "h": 0.0009765625,
        "e_n": 8.400373105454939e-09,
        "beta": -3.9645609742952543
      },
      {
        "n": 2048,
        "h": 0.00048828125,
        "e_n": 1.2195122051084084e-07,
        "beta": -3.859706979635533
      }
    ]
  },
  "dense_study": {
    "description": "L-inf error on a 10x-finer grid via local element basis; order 4 then round-off",
    "table": [
      {
        "n": 32,
        "h": 0.03125,
        "e_star_n": 5.402332782431223e-07,
        "beta": null
      },
      {
        "n": 64,
        "h": 0.015625,
        "e_star_n": 3.4594446804653376e-08,
        "beta": 3.9649701360591583
      },
      {
        "n": 128,
        "h": 0.0078125,
        "e_star_n": 2.1885099189014557e-09,
        "beta": 3.982519643311164
      },
      {
        "n": 256,
        "h": 0.00390625,
        "e_star_n": 1.376106496463837e-10,
        "beta": 3.991284893460215
      },
      {
        "n": 512,
        "h": 0.001953125,
        "e_star_n": 5.380799417142867e-10,
        "beta": -1.967228403610448
      },
      {
        "n": 1024,
        "h": 0.0009765625,
        "e_star_n": 8.400447198964045e-09,
        "beta": -3.964573699190374
      },
      {
        "n": 2048,
        "h": 0.00048828125,
        "e_star_n": 1.2195126741776363e-07,
        "beta": -3.8596948096538584
      }
    ],
    "observed_order_clean_regime": 3.98
  },
  "asymptotic_order": 4,
  "notes": "nodal values superconvergent (~round-off); dense error is the genuine O(h^4) field error and requires the slope DOFs in the reconstruction; round-off (ill-conditioning ~ n^4) dominates for n >= 512.",
  "artifact": {
    "n": 256,
    "points": 2561,
    "dense_max_error": 1.376106496463837e-10
  }
}