📝 SummarXiv

Abstract

Nonlinear PDEs give rise to complex dynamics that are often difficult to analyze in state space due to their relatively large numbers of degrees of freedom, ill-conditioned operators, and changing spatial and parameter resolution requirements. This work introduces ff-bifbox: a new open-source toolbox for performing numerical branch tracing, stability/bifurcation analysis, resolvent analysis, and time integration of large, time-dependent nonlinear PDEs discretized on adaptively refined meshes in two and three spatial dimensions. Spatial discretization is handled using finite elements in FreeFEM, with the discretized operators manipulated in a distributed framework via PETSc. Following a summary of the underlying theory and numerics, results from three examples are presented to validate the implementation and demonstrate its capabilities. The considered examples, which are provided with runnable ff-bifbox code, include: a 3-D Brusselator system, a 3-D plate buckling system, and a 2-D compressible Navier--Stokes system. In addition to reproducing results from prior studies, novel results are presented for each system.