  Partial Differential Equations (PDEs) describe several problems relevant to
many fields of applied sciences, and their discrete counterparts typically
involve the solution of sparse linear systems. In this context, we focus on the
analysis of the computational aspects related to the solution of large and
sparse linear systems with HPC solvers, by considering the performances of
direct and iterative solvers in terms of computational efficiency, scalability,
and numerical accuracy. Our aim is to identify the main criteria to support
application-domain specialists in the selection of the most suitable solvers,
according to the application requirements and available resources. To this end,
we discuss how the numerical solver is affected by the regular/irregular
discretisation of the input domain, the discretisation of the input PDE with
piecewise linear or polynomial basis functions, which generally result in a
higher/lower sparsity of the coefficient matrix, and the choice of different
initial conditions, which are associated with linear systems with multiple
right-hand side terms. Finally, our analysis is independent of the
characteristics of the underlying computational architectures, and provides a
methodological approach that can be applied to different classes of PDEs or
with approximation problems.
