Abstract—Krylov methods play a major role in solving Partial Differential Equations (PDEs) due to their scalability and low memory requirements. However, for difficult problems, Krylov methods exhibit slow convergence and may even not always converge. Increasing the numerical precision can improve the convergence rate of Krylov methods. In the current work, we evaluate the effect of Variable Precision (VP) on two Krylov-based solvers. Our solvers were applied to a relatively difficult PDE, discretized with the Spectral Element Method (SEM), which produces a set of dense and poorly conditioned system of linear equations. We show that, increasing the numerical precision allows us to both speedup the convergence of the solver and more accurately estimate the residual error, making the solver more reliable.

Keywords—Algebraic solver, extended precision, spectral elements

Cite: Alexandre Hoffmann, Yves Durand, Jérôme Fereyre, "Accelerating Spectral Elements Method with Extended Precision: A Case Study," International Journal of Applied Physics and Mathematics, vol. 14, no. 2, pp. 45-58, 2024.

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