  Differential equations are ubiquitous in models of physical phenomena.
Applications like steady-state analysis of heat flow and deflection in elastic
bars often admit to a second order differential equation. In this paper, we
discuss the use of a quantum annealer to solve such differential equations by
recasting a finite element model in the form of an Ising hamiltonian. The
discrete variables involved in the Ising model introduce complications when
defining differential quantities, for instance, gradients involved in
scientific computations of solid and fluid mechanics. To address this issue, a
graph coloring based methodology is proposed which searches iteratively for
solutions in a subspace of weak solutions defined over a graph, hereafter
called as the 'box algorithm.' The box algorithm is demonstrated by solving a
truss mechanics problem on the D-Wave quantum computer.
