A Geometric Local Parameterization Method for Generalized Hele-Shaw Free Boundary Problems with Source Terms

Jul 16, 2026

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A meshfree numerical framework addresses generalized Hele–Shaw free boundary problems with surface tension and general source terms using geometric local parameterization and boundary integral ideas. The pressure is decomposed into a particular solution and a harmonic component, reformulating the system into boundary-only equations and avoiding repeated volume meshing of the evolving domain. For source terms without closed-form particular solutions, an eigenfunction approximation on a fixed domain is proposed with truncation and coefficient error estimates. Numerical tests confirm accuracy and convergence, including an application to a tumor growth model.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages20

Reading Time50 min

Introduction

Background and motivation

Related numerical methods

Boundary integral methods and their limitation with source terms

Proposed approach and contributions

Paper organization

arXiv :2607 .02880v1 [math .NA] 3 Jul 2026 A GEOMETRIC LOCAL PARAMETERIZ ATION METHOD FOR GENERALIZED HELE–SHAW FREE BOUNDARY PROBLEMS WITH SOURCE TERMS A PREPRINT Zengyan Zhang and Wenrui Hao Department of Mathematics The Pennsylvania State University, University Park, PA 16802, USA [zzz5527@psu.edu](zzz5527@psu.edu) and wxh64@psu.edu John Harlim Department of Mathematics, Institute for Computational and Data Sciences The Pennsylvania State University, University Park, PA 16802, USA [jharlim@psu.edu](jharlim@psu.edu) July 7, 2026 ABSTRACT We develop a meshfree numerical framework for Hele–Shaw free boundary problems with surface tension and source terms based on geometric local parameterization and boundary integral methods. By decomposing the pressure into a particular solution and a harmonic component, the problem is reformulated into a boundary-only system, avoiding volumetric meshing of the evolving domain. For general source terms, we propose an eigenfunction-based approximation on a fixed domain and establish error estimates for both the truncation and coefficient approximation. Numerical experiments verify the accuracy and convergence of the proposed method, and an application to a tumor growth model demonstrates its effectiveness for coupled movingboundary problems. 1 Introduction Free boundary problems arise in a broad range of scientific applications, including fluid dynamics, porous media flow, and tumor growth [9, 11, 15] . Among them, the generalized Hele-Shaw problem with surface tension serves asa fundamental model for interface motion driven by pressure gradients under the influence of surface tension [3, 7, 30] . This problem has been extensively studied since the seminal work ofSaffman and Taylor [33], following HeleShaw’s pioneering experimental studies of viscous flow between two closely spaced plates [18] . In its classical form, the pressure satisfies Laplace’s equation in the evolving domain, and the interface velocity is determined by Darcy’slaw through the pressure gradient. Owing to the coupling between the pressure field and the evolving interface, the development of accurate and efficient numerical methods for Hele-Shaw problems remains a challenging task[35] . A variety of numerical methods have been developed for Hele-Shaw free boundary problems and their applications, including finite difference methods, finite element methods, level-set methods, and phase field methods [6, 26, 31, 32] . Although these approaches have been successfully applied to a wide range of free boundary problems, they generally require discretization of the computational domain together with repeated interface reconstruction or mesh generation as the free boundary evolves, leading to increased computational cost. Boundary integral methods provide an attractive alternative for Hele-Shaw problems because they exploit the Green’s function of the governing equation to reduce the problem to the evolving interface, thereby avoiding vol- A PREPRINT-JULY 7, 2026 umetric discretization and reducing the dimensionality of the computation by one [20, 29] . Moreover, rigorous convergence studies of boundary integral methods for free boundary problems have been established [16], providing a solid theoretical foundation for their application. For example, we recently introduced a geometric local parameterization method based on generalized moving least squares (GMLS) for solving classical Hele–Shaw problems directly on point clouds [36] . By reconstructing local manifold parameterizations from unorganized boundary points, the method accurately approximates geometric quantities and boundary integrals without requiring global parameterization or remeshing. Nevertheless, most existing formulations (including [36]) are limited to homogeneous equations or rely on parameterized interfaces and body-fitted meshes. When nonzero source terms are present, as in tumor growth and reactive porous media model

FAQs

What key idea allows the method to avoid volumetric meshing during interface evolution?

The pressure is decomposed into a particular solution and a harmonic component, which reformulates the problem into a boundary-only system, eliminating repeated volume integration over the evolving domain.

How does the framework handle generalized source terms when a closed-form particular solution is unavailable?

It uses an eigenfunction-based approximation on a fixed domain, expanding the particular part via Laplacian eigenfunctions and solving the associated Poisson problem offline.

How is the approach validated, and what application demonstrates its usefulness?

Spatial and temporal convergence studies are carried out through numerical experiments, and the method is applied to a nutrient-driven tumor growth model to illustrate coupled moving-boundary evolution with source terms without repeated volumetric remeshing.

Jul 16, 2026

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