A Parallel Least Squares, Conjugate Gradient, Finite Element Method Solver for Velocity-Current MHD Equations

Due to the symmetry of weak formulations for the Navier-Stokes equations and the velocity-current MHD (magnetohydrodynamics) equations, we propose a least squares formulation and numerical approximation mdethod for the velocity-current MHD equations that is based on work by Roland Glowinski and fellow authors. A parallel, finite element method solver was developed that utilizes the open-source, C++ software library deal.II and wraps into the libraries p4est and Trilinos. A block-diagonal preconditioner is utilized for convergence of the conjugate gradient method. The weak formulation, finite-dimensional approximation, and algorithm implementation are discussed with application interest toward optimization of crystal growth processes.