Solving Partial Differential Equations on Evolving Surfaces via the Constrained Least-Squares and Grid-Based Particle Method

TL;DR

This work tackles the challenge of solving parabolic PDEs on evolving surfaces by marrying the Grid-Based Particle Method (GBPM) with a constrained least-squares ghost sampling points (CLS-GSP) reconstruction. The approach integrates a new local reconstruction weighting to mitigate cross-segment contamination in high-curvature evolution and employs CLS-GSP with both polynomial and radial-basis-function bases to obtain well-conditioned discretizations of surface differential operators such as the Laplace-Beltrami. Key contributions include the soft weighting strategy for local reconstruction, the CLS-GSP framework on GBPM representations, and a unified algorithm coupling surface representation, operator assembly, and time stepping across various test problems. Numerical experiments across oscillating ellipsoids, evolving toruses, tumor-growth surfaces, and Cahn-Hilliard on ellipsoids demonstrate improved accuracy, stability, and robustness, highlighting the method’s potential for broad applications in biology and materials science.

Abstract

We present a framework for solving partial different equations on evolving surfaces. Based on the grid-based particle method (GBPM) [18], the method can naturally resample the surface even under large deformation from the motion law. We introduce a new component in the local reconstruction step of the algorithm and demonstrate numerically that the modification can improve computational accuracy when a large curvature region is developed during evolution. The method also incorporates a recently developed constrained least-squares ghost sample points (CLS-GSP) formulation, which can lead to a better-conditioned discretized matrix for computing some surface differential operators. The proposed framework can incorporate many methods and link various approaches to the same problem. Several numerical experiments are carried out to show the accuracy and effectiveness of the proposed method.

Figures (14)

• Figure 1: Grid Based Particle Method. (a) Initialization, (b) after motion, (c) after re-sampling, (d) after activating new grid points with their footpoints, (e) after inactivating grid points with their footpoints. Blue circles denote active grid points, and red squares denote their corresponding footpoints on the interface. In practice, we do not use such a thick computational tube. We pick it here to show the effect of each step in our algorithm clearly.
• Figure 2: The local reconstruction step involves a construction of the tangent plane using a normal vector $\mathbf{n}$ estimated from the data and a projection step of sampled points collected from a local neighborhood.
• Figure 3: The local samplings of the function. The black dot denotes the target point $\mathbf{x}_0$. The blue squares are the $n$-nearest scattered data points projection onto the local tangent plane at $\mathbf{x}_0$. The red triangles are the ghost sample points surrounding the target point $\mathbf{x}_0$ on the tangent plane.
• Figure 4: (Example \ref{['Ex:MGBPM']}) Motion under a single vortex with the reversible motion using the resolution $100^3$ at $t = 0, 0.375, 0.75$, and 1.5.
• Figure 5: (Example \ref{['Ex:MGBPM']}) (a) The change in the number of footpoints at different times and (b) the error in the radius of the sphere $r-r_0$ at the final time.

Theorems & Definitions (2)

• Remark 1
• Remark 2