MooAFEM and octAFEM3D: Computational study of the smoothed adaptive finite element method (S-AFEM) for second-order elliptic PDEs

We investigate the smoothed adaptive finite element method (S-AFEM) proposed in [Mulita, Giani, Heltai: SIAM J. Sci. Comput. 43, 2021]. S-AFEM modifies the classical adaptive finite element method (AFEM) by performing accurate discrete solves only on periodically determined mesh levels, while the intermediate levels employ a fixed number of cheap smoothing iterations. This strategy generates adapted meshes comparable to those of AFEM at substantially lower computational cost. We consider standard smoothers such as Richardson, Gauss-Seidel, conjugate gradient, and multigrid schemes. The algorithm ensures unconditional full R-linear convergence and, for sufficiently small adaptivity parameters, optimal convergence rates with respect to the overall computational cost for general second-order linear elliptic PDEs. The 2D experiments were conducted using the object-oriented Matlab software package MooAFEM. For the experiments in 3D, we extended the octAFEM3D software package. The numerical experiments reproduce the results reported in the associated publication. They validate the convergence results, analyze runtime performance, and underline the potential of S-AFEM for speed-up in AFEM computations.