Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A hybrid high-order method for the biharmonic problem

Yizhou Liang, Ngoc Tien Tran

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.

A hybrid high-order method for the biharmonic problem

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 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.

Paper Structure

This paper contains 26 sections, 14 theorems, 131 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Discretization
  3. Triangulation
  4. Local ansatz space and reconstructions
  5. Discrete problem
  6. Error analysis of the source problem
  7. Right-inverse
  8. A priori
  9. A posteriori
  10. Biharmonic eigenvalue problem
  11. Discrete eigenvalue problem
  12. Lower eigenvalue bounds
  13. Spectral correctness
  14. A priori
  15. A posteriori
  16. ...and 11 more sections

Key Result

Lemma 2.1

Any $v \in H^2(T)$ satisfies the $L^2$ orthogonality $\nabla^2 (v - R_T I_T v) \perp \nabla^2 P_{k+2}(T)$. In particular, $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 5.1: Sparsity pattern of stiffness matrix for $k = 2$ on uniform mesh with 2305 degrees of freedom
  • Figure 5.2: Convergence history plot of $\eta$ (left) and adaptive triangulation with 445 triangles (right) for the experiment in \ref{['sec:num_ex_bihar']}
  • Figure 5.3: Convergence history plot of $\lambda(1) - \mathrm{LEB}(1)$ (left) and adaptive triangulation with 4610 triangles (right) for the experiment in \ref{['sec:num_ex_eig']}

Theorems & Definitions (38)

  • Lemma 2.1: commuting
  • proof
  • Lemma 2.2: optimality of $s_T$
  • proof
  • Remark 2.3: relation to WG method of LiWangWangZhang2024
  • Remark 2.4: static condensation
  • Remark 2.5: computational cost
  • Theorem 2.6: existence and uniqueness of solutions
  • proof
  • Remark 2.7: alternative side conditions for $R_T$
  • ...and 28 more