Symplectic finite element scheme: application to a driven problem with a regular singularity

Abstract

A new finite element (FE) scheme, based on the decomposition of a second order differential equation into a set of first order symplectic (Hamiltonian) equations, is presented and tested on a one-dimensional, driven SturmdashLiouville problem. Error analysis shows improved cubic convergence in the energy norm for piecewise linear “tent” elements, as compared to quadratic convergence for the standard and symplectic hybrid (i.e. ‘tent’ and piecewise constant) FE methods. The convergence deteriorates in the presence of a regular singular point, but can be recovered by appropriate mesh node packing. Optimal mesh packing exponents are derived to ensure cubic (respectively quadratic for the hybrid FE method) convergence with minimal numerical error. The symplectic hybrid FE scheme is shown to be approximately 30–40 times more accurate than the standard FE scheme, for an exact test problem based on determining the nonideal magnetohydrodynamic stability of a fusion plasma. A further suppression of the error by one order of magnitude is achieved for the symplectic tent element method.