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

Jul 16, 2026

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A hybrid high-order (HHO) finite element approximation is developed and analyzed for a nonlocal nonlinear Kirchhoff-type boundary value problem. The method uses arbitrary-order polynomial approximations on structured and unstructured polytopal meshes. The study proves existence and uniqueness of the discrete solution and derives an optimal-order error estimate in the discrete energy norm. The nonlinear discrete system is solved via Newton iterations and validated through numerical experiments.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages19

Reading Time48 min

Introduction

Model problem

Literature survey

arXiv :2510 . 15574v2 [math .NA] 11 Jul 2026 A Hybrid High-Order Finite Element Method for a Nonlocal Nonlinear Problem of Kirchhoff Type Gouranga Mallik 1 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 arbitraryorder 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’siterations on the sparse matrix system. We perform numerical tests to substantiate the theoretical results. Keywords: Finite element, hybrid high-order, a priori error estimate, Kirchhoff type, nonlocal nonlinear. 1. Introduction 1.1. Model problem We consider a nonlocal nonlinear elliptic boundary value problem of Kirchhoff type: −􀀢 􀀒∫Ω |∇􀁄|2 dx􀀓 Δ􀁄 = 􀀵 (􀁇 , 􀁄) in Ω, 􀁄 = 0 in 􀁭Ω, (1.1) where Ω ⊂ R􀀳 , 􀀳 = 2,3 is a bounded polytopal domain with Lipschitz boundary 􀁭Ω, the Euclidean norm of the gradient, |∇􀁄|2 = ( ~~􀁭􀁭􀁄􀁇~~ ) 2 + ( ~~􀁭􀁭􀁄~~􀁈 ) 2 , the Laplacian operator Δ􀁄 = ~~􀁭~~􀁭2􀁇~~􀁄~~2 + ~~􀁭~~􀁭2􀁈~~􀁄~~2 and the given source term 􀀵 : Ω × R → R and the nonlocal coefficient term 􀀢 : R+ → R. Equation (1.1) involves an integral over Ω, implying that the problem is a nonlocal and does not hold pointwise. The main aim of the article is to design and analyze hybrid high-order polytopal finite element approximation for the Kirchhoff type problem (1.1) . We establish the existence and uniqueness of the discrete problem and derive an error estimate. 1.2. Literature survey The above problem (1.1) corresponds to the stationary form of the generalized Kirchhoff equation: 􀁭􀁭 − 􀀢 􀀒∫Ω |∇􀁄|2 dx􀀓 Δ􀁄 = 􀀵 (􀁇 , 􀁄) , (1.2) 1Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore-632014, India. [Email. gouranga.mallik@vit.ac.in](Email. gouranga.mallik@vit.ac.in). which is an extension of the classical wave equation that accounts for volume changes in a vibrating body. In [11, 34], problems structurally related to (1.1), where the nonlocal term is of the form 􀀢 ( ∫Ω 􀁄 dx), model certain biological phenomena of the density of bacteria in the domain Ω . The well-posedness and qualitative properties of the problems (1.1) and (1.2) are studied in [16, 4]; see also the survey article [39] . During the past two decades, the numerical approximation of Kirchhoff-type problems of the forms (1.1) and (1.2) has become a growing area of research. In [44], Peradze analyzed the one-dimensional Kirchhoff string problem using a spectral method. A seminal contribution by Gudi [34] to the finite element approximation of the Kirchhoff type problem (1.1) involves areformulation into an equivalent discrete system, in which the Jacobian arising from the linearization preserves sparsity. The author established the existence and uniqueness of the discrete solution, together with optimal-order error estimates in the energy norm. In [26], the authors discuss a priori and a posteriori error estimates of conforming and mixed FEMs for the nonlocal Kirchhoff problem of type (2 .3) . In [1], a virtual element method for (1 . 1) is considered in a global 􀀝1-conforming virtual element space. Recently, in [43], the authors consider an a posteriori error estimate for (1.1) and prove the convergence of the adaptive FEM. Over the past decades, significant progress has been made in developing high-order polynomial approximation techniques for partial differential equations on general polytopal meshes. Among these, several well-established approaches have been developed and refined. The Hybridizable Discontinuous Galerkin (HDG) method was proposed in [15] . The Virtual Element Method (VEM) has been extensively investigated in [5, 6, 7] . Other notable frameworks include the

FAQs

What problem does the hybrid HHO method target?

It targets a nonlocal nonlinear elliptic boundary value problem of Kirchhoff type, where the governing equation contains an integral term over the domain, making it nonlocal.

What theoretical results are established for the discrete problem?

The article proves existence and uniqueness of the discrete solution and derives an optimal-order error estimate in the discrete energy norm.

How is the resulting nonlinear discrete system solved and verified?

Newton iterations are applied to the sparse matrix system, and numerical tests are carried out to substantiate the theoretical convergence and error estimates.

Jul 16, 2026

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