SYMPLECTIC FINITE-ELEMENT SCHEME - APPLICATION TO A DRIVEN PROBLEM WITH A REGULAR SINGULARITY

A new finite element (FE) scheme, based on the decomposition of a seco nd order differential equation into a set of first order symplectic (H amiltonian) equations, is presented and tested on a one-dimensional, d riven Sturm-Liouville problem. Error analysis shows improved cubic con vergence in the energy norm for piecewise linear ''tent'' elements, as compared to quadratic convergence for the standard and symplectic hyb rid (i.e. 'tent' and piecewise constant) FE methods. The convergence d eteriorates in the presence of a regular singular point, but can be re covered by appropriate mesh node packing. Optimal mesh packing exponen ts are derived to ensure cubic (respectively quadratic for the hybrid FE method) convergence with minimal numerical error The symplectic hyb rid FE scheme is shown to be approximately 30-40 times more accurate t han the standard FE scheme, for an exact test problem based on determi ning 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.