Finite element method for nonlocal problems of Kirchhoff-type in domains with moving boundary

Abstract

This paper is devoted to the analysis of the finite element method for the mixed problem for the Kirchhoff nonlinear model given by the hyperbolic-parabolic equations in a bounded noncylindrical domain with moving boundaries. With the use of the coordinate transformation which fixes the boundaries, the semidiscrete formulation is presented and the convergence and error bounds in the energy norm and for the first order derivative with respect to time in the L2-norm are established. In particular, the error in the energy norm and for the first order derivative with respect to time in the L2-norm is shown to converge with the optimal order O(hr) with respect to the mesh size h and the polynomial degree r≥1. To obtain the fully discrete solution, the generalized-α method is adapted to the semidiscrete formulation. The test problems used are designed to illustrate the behavior of the algorithms. Some numerical tests using Matlab are performed to confirm our theoretical findings.