  In this paper we aim to combine tools from variational calculus with modern
techniques from quaternionic analysis that involve Dirac type operators and
related hypercomplex integral operators. The aim is to develop new methods for
showing geometry independent explicit global existence and uniqueness criteria
as well as new computational methods with special focus to the stationary
incompressible viscous magnetohydrodynamic equations.
  We first show how to specifically apply variational calculus in the
quaternionic setting. To this end we explain how the mountain pass theorem can
be successfully applied to guarantee the existence of (weak) solutions. To
achieve this, the quaternionic integral operator calculus serves as a key
ingredient allowing us to apply Schauder's fixed point theorem. The advantage
of the approach using Schauder's fixed point theorem is that it is also
applicable to large data since it does not require any kind of contraction
property.
  These consideration will allow us to provide explicit iterative algorithms
for its numerical solution. Finally to obtain more precise a-priori estimates
one can use in the situations dealing with small data the Banach fixed point
theorem which then also grants the uniqueness.
