arXiv subjects

Arum Lee

Publications and source records attributed to Arum Lee.

3 recordsLinked to original sources

arXiv2026

Weak Solutions and Inertial Limits for Quasi-static Filtrations

A quasi-static filtration system, comprising a poroelastic solid coupled to an incompressible free-flow, is considered in 3D. Across a flat 2D interface, the Beavers-Joseph-Saffman coupling conditions are taken. The system constitutes a doubly elliptic-parabolic coupling and can be seen as a degenerate case of the inertial Biot-Stokes dynamics. These dynamics cannot be easily recovered by simply sending the inertial parameters to zero; however, inserting a viscoelastic regularization of the inertial Biot system allows us to construct weak solutions in the inertial limit. Subsequently, we can pass to the limit in the regularization parameter to obtain quasi-static weak solutions. This addresses an open singular/degenerate limiting problem in the class of filtration models, and permits future analyses of uniqueness and regularity of solutions. This linear construction also provides a foundation for the later incorporation of physically-motivated nonlinear poroelastic effects via fixed point methods.

math.AP↗

arXiv2024

A shape derivative algorithm for reconstructing elastic dislocations in geophysics

We consider the inverse problem of determining an elastic dislocation that models a seismic fault in the quasi-static regime of aseismic, creeping faults, from displacement measurements made at the surface of Earth. We derive both a distributed as well as a boundary shape derivative that encodes the change in a misfit functional between the measured and the computed surface displacement under infinitesimal movements of the dislocation and infinitesimal changes in the slip vector, which gives the displacement jump across the dislocation. We employ the shape derivative in an iterative reconstruction algorithm. We present some numerical test of the reconstruction algorithm in a simplified 2D setting.

math.AP↗

arXiv2019

A regularization approach for solving Poisson's equation with singular charge sources and diffuse interfaces

Singular charge sources in terms of Dirac delta functions present a well-known numerical challenge for solving Poisson's equation. For a sharp interface between inhomogeneous media, singular charges could be analytically treated by fundamental solutions or regularization methods. However, no analytical treatment is known in the literature in case of a diffuse interface of complex shape. This letter reports the first such regularization method that represents the Coulomb potential component analytically by Green's functions to account for singular charges. The other component, i.e., the reaction field potential, then satisfies a regularized Poisson equation with a smooth source and the original elliptic operator. The regularized equation can then be simply solved by any numerical method. For a spherical domain with diffuse interface, the proposed regularization method is numerically validated and compared with a semi-analytical quasi-harmonic method.

math.NA↗