A Unified Isoparametric Moving Mesh PDE Method for P1 and Q1 Elements with Certified Local Nonsingularity
Abstract
We develop a unified isoparametric moving mesh partial differential equation (MMPDE) method for P1 simplicial and Q1 tensor-product elements, including elements on embedded surfaces. Mesh movement follows a prescribed symmetric positive-definite metric through equidistribution and alignment. For Q1 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.
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BibTeX
@misc{li2026unified,
title = {{A Unified Isoparametric Moving Mesh PDE Method for P1 and Q1 Elements with Certified Local Nonsingularity}},
author = {Xianping Li},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05026},
eprint = {2611.05026},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05026},
}
RIS
TY - UNPB
TI - A Unified Isoparametric Moving Mesh PDE Method for P1 and Q1 Elements with Certified Local Nonsingularity
AU - Xianping Li
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05026
UR - https://arxiv.org/abs/2611.05026
AB - 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.
ER -