An Implementation of the Finite Element Method for the Velocity-Current Magnetohydrodynamics Equations

Abstract

The aim of this research is to extend the numerical results in [78] of a velocity-current magnetohydrodynamics formulation proposed by A.J. Meir and Paul G. Schmidt in [74] for a stationary flow that models a conductive fluid in a bounded domain. A parallel finite element algorithm was successfully implemented on a high-performance computing, distributed-memory architecture at the Alabama Supercomputer Center (ASC) using the freely available, open-source academic and government libraries deal.ii, p4est and Trilinos. Extending the work of Elman, Silvester and Wathen [37] for the Navier-Stokes equations, a Schur complement preconditioner was developed for the current saddle-point problem to successfully utilize the iterative Krylov subspace solver GMRES (generalized minimal residual method) and solve large linear systems of equations arising from mesh refinement. To simplify and lower operation costs in forming the preconditioner, spectral equivalence was established between the Schur complement and a mass matrix. The resulting C++ code was tested succesfully on problems from [78] with similar results.