Conservative Finite Element Methods for Kirchhoff-Type Equations

Abstract

We construct and analyse a fully discrete scheme for a Kirchhoff-type nonlinear integro-differential equation in two spatial dimensions with time-dependent coefficients. The scheme combines a symmetric three-layer time discretization of Crank--Nicolson type with a conforming finite element method of arbitrary polynomial degree p and a Picard iteration for the nonlinearity. We prove that the Picard iteration is well defined and, that the iterates converge to the Ritz projection of the exact solution measured in fractional-power norms of the discrete Laplacian. Running the same energy argument in the unshifted norms, and under the same hypotheses, we then obtain the optimal order 𝒪 ( h^p+1 + τ² ) in L² and 𝒪 ( h^p + τ² ) in H¹ for the full error. For the homogeneous problem with constant coefficients, the exact solution of the scheme moreover preserves a discrete energy identically. Numerical experiments with manufactured solutions reproduce the predicted second order in time and the full spatial order h^p+1 in the L²-norm.

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BibTeX

@misc{dorich2026conservative,
  title = {{Conservative Finite Element Methods for Kirchhoff-Type Equations}},
  author = {Benjamin Dörich and Felix Krumbiegel and Jemal Rogava and Andreas Rupp and Zurab Vashakidze},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.31487},
  eprint = {2610.31487},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.31487},
}

RIS

TY  - UNPB
TI  - Conservative Finite Element Methods for Kirchhoff-Type Equations
AU  - Benjamin Dörich
AU  - Felix Krumbiegel
AU  - Jemal Rogava
AU  - Andreas Rupp
AU  - Zurab Vashakidze
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.31487
UR  - https://arxiv.org/abs/2610.31487
AB  - We construct and analyse a fully discrete scheme for a Kirchhoff-type nonlinear integro-differential equation in two spatial dimensions with time-dependent coefficients. The scheme combines a symmetric three-layer time discretization of Crank--Nicolson type with a conforming finite element method of arbitrary polynomial degree p and a Picard iteration for the nonlinearity. We prove that the Picard iteration is well defined and, that the iterates converge to the Ritz projection of the exact solution measured in fractional-power norms of the discrete Laplacian. Running the same energy argument in the unshifted norms, and under the same hypotheses, we then obtain the optimal order 𝒪 ( h^p+1 + τ^2 ) in L^2 and 𝒪 ( h^p + τ^2 ) in H^1 for the full error. For the homogeneous problem with constant coefficients, the exact solution of the scheme moreover preserves a discrete energy identically. Numerical experiments with manufactured solutions reproduce the predicted second order in time and the full spatial order h^p+1 in the L^2-norm.
ER  -

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