A Unified Isoparametric Moving Mesh PDE Method for P1 and Q1 Elements with Certified Local Nonsingularity

We develop a unified isoparametric moving mesh partial differential equation (MMPDE) method for $P_1$ simplicial and $Q_1$ tensor-product elements, including elements on embedded surfaces. Mesh movement follows a prescribed symmetric positive-definite metric through equidistribution and alignment. For $Q_1$ elements, we prove that bounded, exactly integrated adaptation energy prevents local Jacobian rank loss on a fixed finite mesh at sharp critical exponents. The result also gives local full-rank preservation for embedded bilinear surface elements. A complete physical-coordinate gradient accounts for metric variation at moving quadrature points. For the discrete MMPDE, we use whole-element validity tests and develop sufficient Bernstein certificates for trilinear hexahedra. An explicit positive-margin bound on each fixed stage is provided by an optional certificate barrier. Numerical examples illustrate metric-driven movement on one-dimensional, planar, volume, and graph-surface meshes. Limitations of pointwise hexahedral Jacobian sampling are also presented.