A comparison of preconditioned iterative solvers for high-order mimetic difference approximations to the Poisson equation

Abstract

Mimetic difference approximations are of increasing interest in scientific computing for their ability to preserve key properties of continuous problems, such as conservation laws, while maintaining a consistent order of accuracy in the interior of the domain and near the boundary. However, their application leads to sparse, locally dense, and potentially ill-conditioned linear systems that are challenging to solve. Using the two-dimensional Poisson equation as the model problem, this thesis evaluates the computational performance of several preconditioned Krylov subspace methods for solving the resulting linear systems. Numerical experiments show that combining a Jacobi preconditioner with augmented GMRES(m) variants significantly reduces the execution times and number of iterations. The results highlight the potential of augmented iterative methods with preconditioning as a robust alternative for large-scale simulations in science and engineering that use mimetic discretizations.