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<p>In this paper, we propose three unisolvent weighted quadratic enrichments to extend classical linear histopolation on 3D tetrahedral meshes. The first combines face and interior weighted moments, the second relies exclusively on volumetric quadratic moments, and the third employs edge supported probabilistic moments. In all three cases, the degrees of freedom are defined by weighted integral functionals associated with suitable probability densities and quadratic trial spaces. We establish the unisolvence of the enriched schemes and derive explicit conditions under which this property holds. Representative density families, including two-parameter symmetric Dirichlet laws and volumetric families defined by convex combinations of densities, are examined in detail, and a general procedure for constructing the associated quadratic basis functions is presented. For the considered parametric density families, the corresponding parameters are selected via a grid search procedure. Numerical experiments show improved accuracy compared to the classical linear histopolation scheme.</p>

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Quadratic weighted histopolation on tetrahedral meshes with probabilistic degrees of freedom

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月26日

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  • Allal Guessab,
  • Federico Nudo

摘要

In this paper, we propose three unisolvent weighted quadratic enrichments to extend classical linear histopolation on 3D tetrahedral meshes. The first combines face and interior weighted moments, the second relies exclusively on volumetric quadratic moments, and the third employs edge supported probabilistic moments. In all three cases, the degrees of freedom are defined by weighted integral functionals associated with suitable probability densities and quadratic trial spaces. We establish the unisolvence of the enriched schemes and derive explicit conditions under which this property holds. Representative density families, including two-parameter symmetric Dirichlet laws and volumetric families defined by convex combinations of densities, are examined in detail, and a general procedure for constructing the associated quadratic basis functions is presented. For the considered parametric density families, the corresponding parameters are selected via a grid search procedure. Numerical experiments show improved accuracy compared to the classical linear histopolation scheme.