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Partial Differential Equations (PDEs) describe several problems relevant tomany fields of applied sciences, and their discrete counterparts typicallyinvolve the solution of sparse linear systems. In this context, we focus on theanalysis of the computational aspects related to the solution of large andsparse linear systems with HPC solvers, by considering the performances ofdirect and iterative solvers in terms of computational efficiency, scalability,and numerical accuracy. Our aim is to identify the main criteria to supportapplication-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/irregulardiscretisation of the input domain, the discretisation of the input PDE withpiecewise linear or polynomial basis functions, which generally result in ahigher/lower sparsity of the coefficient matrix, and the choice of differentinitial conditions, which are associated with linear systems with multipleright-hand side terms. Finally, our analysis is independent of thecharacteristics of the underlying computational architectures, and provides amethodological approach that can be applied to different classes of PDEs orwith approximation problems.

Evaluating Accuracy and Efficiency of HPC Solvers for Sparse Linear Systems with Applications to PDEs