Search

In this paper, a novel energy-preserving numerical scheme for nonlinear Hamiltonian wave equations with Neumann boundary conditions is proposed and analyzed based on the blend of spatial discretization by finite element method (FEM) and time discretization by Average Vector Field (AVF) approach. We first use the finite element discretization in space, which leads to a system of Hamiltonian ODEs whose Hamiltonian can be thought of as the semi-discrete energy of the original continuous system. The stability of the semi-discrete finite element scheme is analyzed. We then apply the AVF approach to the Hamiltonian ODEs to yield a new and efficient fully discrete scheme, which can preserve exactly (machine precision) the semi-discrete energy. The blend of FEM and AVF approach derives a new and efficient numerical scheme for nonlinear Hamiltonian wave equations. The numerical results on a single-soliton problem and a sine-Gordon equation are presented to demonstrate the remarkable energy-preserving property of the proposed numerical scheme. [ABSTRACT FROM AUTHOR]

Domain decomposition techniques provide powerful tools for the numerical approximation of partial differential equations. In this paper, we analyze the Nitsche method for the Lamé operator, establish a priori error estimates and compare this method with the mortar method using dual Lagrange multiplier spaces. Both methods can be applied to non-matching triangulations. We use a multigrid algorithm to solve the algebraic systems. Although we have a mesh dependent bilinear form, optimal-cycle convergence rates can be obtained. Numerical results for the two methods with linear and quadratic finite elements illustrate the performance and flexibility of these nonconforming discretization techniques. [ABSTRACT FROM AUTHOR]