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A hybrid high-order method for the biharmonic problem

· 2025 · math.NA · arXiv 2504.16608

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abstract

This paper proposes a new hybrid high-order discretization for the biharmonic problem and the corresponding eigenvalue problem. The discrete ansatz space includes degrees of freedom in $n-2$ dimensional submanifolds (e.g., nodal values in 2D and edge values in 3D), in addition to the typical degrees of freedom in the mesh and on the hyperfaces in the HHO literature. This approach enables the characteristic commuting property of the hybrid high-order methodology in any space dimension. The main results are guaranteed lower eigenvalue bounds of higher order. Furthermore, we derive quasi-best approximation estimates as well as reliable and efficient a~posteriori error estimators under minimal regularity assumptions on the exact solution. The latter motivates an adaptive mesh-refining algorithm that empirically recovers optimal convergence rates for singular solutions.

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math.NA 2

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2026 2

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representative citing papers

An error analysis of discrete Kirchhoff elements

math.NA · 2026-06-30 · unverdicted · novelty 6.0 · 2 refs

Error analysis of DKT elements for biharmonic equation gives regularity-free first-order convergence bounds via best-approximation and oscillation terms, plus 3D extension and a posteriori estimates.

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  • An error analysis of discrete Kirchhoff elements math.NA · 2026-06-30 · unverdicted · none · ref 29 · 2 links · internal anchor

    Error analysis of DKT elements for biharmonic equation gives regularity-free first-order convergence bounds via best-approximation and oscillation terms, plus 3D extension and a posteriori estimates.

  • Quasi-optimal polytopal finite element methods for biharmonic equation math.NA · 2026-05-20 · unverdicted · none · ref 21 · internal anchor

    Quasi-optimal and lower-order error estimates are established for WG, DG, and HHO methods for the biharmonic equation on polytopal meshes with minimal regularity, plus efficient stabilization in a posteriori estimators.