  It is well-known that the standard Galerkin formulation, which is often the
formulation of choice under the finite element method for solving self-adjoint
diffusion equations, does not meet maximum principles and the non-negative
constraint for anisotropic diffusion equations. Recently, optimization-based
methodologies that satisfy maximum principles and the non-negative constraint
for steady-state and transient diffusion-type equations have been proposed. To
date, these methodologies have been tested only on small-scale academic
problems. The purpose of this paper is to systematically study the performance
of the non-negative methodology in the context of high performance computing
(HPC). PETSc and TAO libraries are, respectively, used for the parallel
environment and optimization solvers. For large-scale problems, it is important
for computational scientists to understand the computational performance of
current algorithms available in these scientific libraries. The numerical
experiments are conducted on the state-of-the-art HPC systems, and a
single-core performance model is used to better characterize the efficiency of
the solvers. Our studies indicate that the proposed non-negative computational
framework for diffusion-type equations exhibits excellent strong scaling for
real-world large-scale problems.
