Adaptive LGMRES Method for High-Order Mimetic Difference Approximaations: Accelerating Convergence of the Poisson Equation

Solving linear systems derived from high-order mimetic finite difference discretizations of partial differenital equaions, e.g. the Poisson equation, presents a significant computational challenge. While these methods preserve essential vector calculus identities in the discrete realm, the resulting matrices are non-symmetric and often ill-conditioned. As the accuracy order increases or the mesh is refined, standard GMRES methods frequently suffer from stagnating convergence rates. In this work, we propose PD-LGMRES, an adaptive variant designed to overcome these limitations through two integrated strategies. First, it utilizes search space enrichment by incorporating correction vectors from previous cycles to preserve relevant directions that would otherwise be lost during restarts. Second, it introduces a dynamic restart adjustment where the subspace size is updated according to the recent behavior of the residual norm. This is governed by a proportional-derivative control law that helps the method avoid stagnation when residual reduction is poor, while maintaining computational efficiency during stable convergence. Numerical experiments were conducted using benchmark problems from the the Mimetic Operator Library Enhanced (MOLE), which provides high-order matricial analogs of differential operators. PD-LGMRES provides a more responsive and stable numerical framework than static methods, effectively handling the convergence degradation typically seen in refined mimetic discretizations.