标题： ff-bifbox：用于非线性PDE分岔分析的可扩展开源工具箱 显示英文标题

标题： ff-bifbox: A scalable, open-source toolbox for bifurcation analysis of nonlinear PDEs

摘要： 非线性偏微分方程（PDEs）会产生复杂的动力学行为，由于其自由度数量相对较大、条件不佳的算子以及不断变化的空间和参数分辨率需求，通常在状态空间中难以分析。 这项工作引入了ff-bifbox：一个用于在二维和三维空间中自适应细化网格上对大规模、时间依赖的非线性偏微分方程进行数值分支追踪、稳定性/分岔分析、求逆分析和时间积分的新开源工具箱。 空间离散化使用FreeFEM中的有限元方法处理，离散化后的算子通过PETSc在一个分布式框架中进行操作。 在总结了基础理论和数值方法之后，通过三个示例的结果来验证实现并展示其能力。 所提供的可运行ff-bifbox代码包括：一个三维Brusselator系统、一个三维板屈曲系统和一个二维可压缩Navier-Stokes系统。 除了重现先前研究的结果外，每个系统还展示了新的结果。 显示英文摘要

摘要： 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.