Conservative Finite Element Methods for Kirchhoff-Type Equations

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 $\mathcal{O}(h^{p+1}+{\tau }^2)$ in $L^2$ and $\mathcal{O}(h^p+{\tau }^2)$ 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.