Meshing method to build a centrosymmetric matrix to solve partial differential equations on an irreducible domain including a planar symmetry

TL;DR

The paper tackles the high computational cost of solving PDEs on irreducible domains by exploiting planar symmetry to produce a centrosymmetric discretization matrix. It introduces a meshing strategy that pairs points across a symmetry plane $\Pi$ and numbers nodes centrosymmetrically, ensuring symmetry in the discretized operator and boundary conditions, which enables a twofold reduction in inversion time and memory. A detailed 3D Poisson example demonstrates how a centrosymmetric mesh yields a centrosymmetric matrix, in contrast to the non-symmetric result from classical numbering, illustrating the practical gain. The authors also provide mathematical expressions showing how a centrosymmetric matrix can be represented with rank-$N$ blocks and how its inverse can be computed efficiently, improving scalability from $O(N'^2)$ to $O(2N^2)$. Overall, the method offers a general, symmetry-based route to accelerate PDE solvers in both FDM and FEM contexts when planar symmetry is present.

Abstract

A general method to generate a centrosymmetric matrix associated with the solving of partial differential equation (PDE) on an irreducible domain by means of a linear equation system is proposed. The method applies to any PDE for which both the domain to solve and the boundary condition (BC) type accept a planar symmetry, while no conditions are required on the BC values and the PDE right hand size function. It is applicable to finite element or finite difference method (FDM). It relies both on the specific construction of a mesh having a planar symmetry and a centrosymmetric numbering of the mesh nodes used to solve the PDE on the domain. The method is exemplified with a simple PDE using FDM. The method allows to reduce the numerical problem size to solve by a factor of two, decreasing as much the computing time and the need of computer memory.

Figures (4)

• Figure 1: Example of a domain reduction using known internal symmetries.
• Figure 2: Example of a reduced domain $\Omega'$ and boundary $\delta \Omega'$ with $\mathcal{N}$ and $\mathcal{D}$ BC still presenting an unexploited planar geometrical symmetry (plane $\Pi$, red).
• Figure 3: a) Classical meshing numbering for the example treated. b) Centrosymmetric meshing numbering as proposed in this work. The plane of symmetry position (perpendicular to $z$) is indicated by a dashed red line.
• Figure 4: a) Classical filling of matrix A. b) Centrosymmetric filling of matrix A. The matrix indexes are colored by value for visual convenience. The symmetry plane is indicated with a red line for b).