Adaptive mesh movement — the MMPDE approach and its applications

A class of moving mesh algorithms based upon a so-called moving mesh partial differential equation (MMPDE) is reviewed. Various forms for the MMPDE are presented for both the simple one- and the higher-dimensional cases. Additional practical features such as mesh movement on the boundary, selection of the monitor function, and smoothing of the monitor function are addressed. The overall discretization and solution procedure, including for unstructured meshes, are briefly covered. Finally, we discuss some physical applications suitable for MMPDE techniques and some challenges facing MMPDE methods in the future.