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A Hybrid High-Order Finite Element Method for a Nonlocal Nonlinear Problem of Kirchhoff Type

Gouranga Mallik

TL;DR

This work develops a Hybrid High-Order (HHO) finite element method for a nonlocal nonlinear Kirchhoff-type problem and proves the existence and uniqueness of the discrete solution along with an optimal-order a priori error estimate in the discrete energy norm. By leveraging arbitrary-order polynomials on general polytopal meshes and local polynomial reconstructions, the method achieves high-order accuracy while enabling static condensation to preserve sparse Jacobians, and a modified formulation with a scalar $d$ ensures a robust Newton solver. The analysis requires mild regularity and a small-mesh condition to control nonlocality and nonlinearity, and it provides rigorous convergence guarantees. Numerical experiments on four polygonal mesh families ($k=0,1,2$) demonstrate robust performance and confirm the theoretical rates, highlighting the approach’s practical relevance for complex geometries.

Abstract

In this article, we design and analyze a hybrid high-order (HHO) finite element approximation for the solution of a nonlocal nonlinear problem of Kirchhoff type. The HHO method involves arbitrary-order polynomial approximations on structured and unstructured polytopal meshes. We establish the existence of a unique discrete solution to the nonlocal nonlinear discrete problem. We derive an optimal-order error estimate in the discrete energy norm. The discrete system is solved using Newton's iterations on the sparse matrix system. We perform numerical tests to substantiate the theoretical results.

A Hybrid High-Order Finite Element Method for a Nonlocal Nonlinear Problem of Kirchhoff Type

TL;DR

This work develops a Hybrid High-Order (HHO) finite element method for a nonlocal nonlinear Kirchhoff-type problem and proves the existence and uniqueness of the discrete solution along with an optimal-order a priori error estimate in the discrete energy norm. By leveraging arbitrary-order polynomials on general polytopal meshes and local polynomial reconstructions, the method achieves high-order accuracy while enabling static condensation to preserve sparse Jacobians, and a modified formulation with a scalar ensures a robust Newton solver. The analysis requires mild regularity and a small-mesh condition to control nonlocality and nonlinearity, and it provides rigorous convergence guarantees. Numerical experiments on four polygonal mesh families () demonstrate robust performance and confirm the theoretical rates, highlighting the approach’s practical relevance for complex geometries.

Abstract

In this article, we design and analyze a hybrid high-order (HHO) finite element approximation for the solution of a nonlocal nonlinear problem of Kirchhoff type. The HHO method involves arbitrary-order polynomial approximations on structured and unstructured polytopal meshes. We establish the existence of a unique discrete solution to the nonlocal nonlinear discrete problem. We derive an optimal-order error estimate in the discrete energy norm. The discrete system is solved using Newton's iterations on the sparse matrix system. We perform numerical tests to substantiate the theoretical results.
Paper Structure (14 sections, 7 theorems, 58 equations, 2 figures, 4 tables)

This paper contains 14 sections, 7 theorems, 58 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Model problem
  3. Literature survey
  4. Contributions
  5. Structure of the paper
  6. Continuous problem and week formulation
  7. Hybrid High-Order discretization
  8. Discrete setting
  9. Discrete spaces
  10. Local reconstructions
  11. Discrete HHO formulation
  12. Error estimate
  13. Numerical Experiments
  14. Conclusion

Key Result

Theorem 2.1

Gudi12_Kirchhof_apriori Assume that (A1)-(A2) hold. Then problem weak_kirchhoff has at least one solution in $H^1_0(\Omega)$. Moreover, any solution of weak_kirchhoff satisfies where $R_1$ is defined in def_R1. In addition, if (A3) holds, then the solution to weak_kirchhoff is unique.

Figures (2)

  • Figure 1: (a) Triangular, (b) Cartesian, (c) hexagonal and (d) Kershaw initial meshes.
  • Figure 2: Convergence histories for the relative discrete $H^1$-error on the (a) triangular, (b) Cartesian, (c) hexagonal and (d) Kershaw meshes.

Theorems & Definitions (10)

  • Theorem 2.1
  • Lemma 3.1: Approximation properties of $R_T^{k+1}\underline{I}_T^k$
  • Theorem 4.1
  • Proposition 4.2
  • Lemma 4.3
  • Theorem 4.4
  • proof
  • Theorem 5.1
  • proof
  • Example 6.1