Symmetry Inheritance and Symmetry-Reduced Finite Element Analysis for Second-Order Linear Elliptic Problems with Robin Boundary Conditions
Abstract
A symmetry framework is developed for second-order linear elliptic equations subject to Robin boundary conditions on bounded domains. Orthogonal transformations of the domain are represented through left group actions on scalar, vector, and second-order tensor fields. The corresponding transformation rules for the gradient, divergence, diffusion flux, and conormal boundary term are derived in detail. Based on these relations, symmetry groups are introduced for the principal coefficient tensor, the first-order and zeroth-order coefficients, the differential operator, the Robin coefficient, and the boundary operator. The symmetry properties of the volume and boundary source terms are then incorporated into a common symmetry group for the complete boundary value problem. Under the assumption of unique solvability, it is proved that every element of this common group is also a symmetry of the solution. For reflection symmetries, homogeneous generalized Neumann conditions are obtained on artificial symmetry boundaries, leading to an exact reduction of the computational domain. A corresponding finite element formulation is presented, and numerical results are provided to verify the theoretical symmetry properties and the validity of the resulting domain-reduction strategy. Three examples illustrate radial reduction from multiple dimensions to one dimension, reflection-based domain reduction, and a variable-coefficient problem in which all coefficients and source terms are nonzero.
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BibTeX
@misc{jiang2026symmetry,
title = {{Symmetry Inheritance and Symmetry-Reduced Finite Element Analysis for Second-Order Linear Elliptic Problems with Robin Boundary Conditions}},
author = {Wei Jiang and Xianlong Pan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01107},
eprint = {2609.01107},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01107},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Symmetry Inheritance and Symmetry-Reduced Finite Element Analysis for Second-Order Linear Elliptic Problems with Robin Boundary Conditions
AU - Wei Jiang
AU - Xianlong Pan
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01107
UR - https://arxiv.org/abs/2609.01107
AB - A symmetry framework is developed for second-order linear elliptic equations subject to Robin boundary conditions on bounded domains. Orthogonal transformations of the domain are represented through left group actions on scalar, vector, and second-order tensor fields. The corresponding transformation rules for the gradient, divergence, diffusion flux, and conormal boundary term are derived in detail. Based on these relations, symmetry groups are introduced for the principal coefficient tensor, the first-order and zeroth-order coefficients, the differential operator, the Robin coefficient, and the boundary operator. The symmetry properties of the volume and boundary source terms are then incorporated into a common symmetry group for the complete boundary value problem. Under the assumption of unique solvability, it is proved that every element of this common group is also a symmetry of the solution. For reflection symmetries, homogeneous generalized Neumann conditions are obtained on artificial symmetry boundaries, leading to an exact reduction of the computational domain. A corresponding finite element formulation is presented, and numerical results are provided to verify the theoretical symmetry properties and the validity of the resulting domain-reduction strategy. Three examples illustrate radial reduction from multiple dimensions to one dimension, reflection-based domain reduction, and a variable-coefficient problem in which all coefficients and source terms are nonzero.
ER -