Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.32066

Optimal convergence analysis of arbitrary Lagrangian-Eulerian finite element methods for two-phase Stokes flow problems with moving interface

Yi Liang·Cheng Wang·Pengtao Sun·Yan Chen·Jiarui Han
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, an arbitrary Lagrangian-Eulerian (ALE)-based finite element method (FEM) is developed and studied in a monolithic framework for a class of two-phase Stokes flow problems with moving interfaces and jump coefficients, where the mixed finite element approximation to Stokes moving interface problems is established and analyzed in both semi- and fully discrete schemes based on the ALE formulation. The key analytical technique involves a specific H^1-projection associated with the ALE-induced mesh motion due to the evolving interface. Optimal convergence properties of the proposed H^1-projection and its ALE temporal derivative are proved in both H^1 and L^2 norms, with which optimal error estimates are obtained for both semi- and fully discrete mixed finite element approximations to the studied Stokes moving interface problem in both H1 and L2 norms as well. Numerical experiments are carried out to validate all derived theoretical results. The developed analytical approach can be extended to Taylor-Hood and MINI mixed elements, as well as to more general two-phase flow problems.

Explore related subjects

math.NAcs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yi Liang, Cheng Wang, Pengtao Sun, Yan Chen, Jiarui Han. 2026-09-25. Optimal convergence analysis of arbitrary Lagrangian-Eulerian finite element methods for two-phase Stokes flow problems with moving interface. https://arxiv.org/abs/2609.32066

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On Trimming Tensor-structured Measurements and Efficient Low-rank Tensor Recovery

In this paper, we take a step towards developing efficient hard thresholding methods for low-rank tensor recovery from memory-efficient linear measurements with tensorial structure. Theoretical guarantees for many standard iterative low-rank recovery methods, such as iterative hard thresholding (IHT), are based on model assumptions on the measurement operator, like the restricted isometry property (RIP). However, tensor-structured random linear maps -- while memory-efficient and convenient to apply -- lack good restricted isometry properties; that is, they do not preserve the norms of low-rank tensors sufficiently well. To address this, we propose local trimming techniques that provably restore point-wise geometry-preservation properties of tensor-structured maps, making them comparable to those of unstructured sub-Gaussian measurements. Then, we propose two novel versions of tensor IHT algorithms: an adaptive gradient trimming algorithm and a randomized Kaczmarz-based IHT algorithm, that efficiently recover low-rank tensors from linear measurements. We provide initial theoretical guarantees for the proposed methods and present numerical experiments on real and synthetic data, highlighting their efficiency over the original TensorIHT for low HOSVD and CP-rank tensors.

math.NA↗
arXiv2026

Corrected Trapezoidal Rules for Near-Singular Surface Integrals Applied to 3D Ellipsoids in Stokes Flow

Interfacial Stokes flow can be efficiently computed using the Boundary Integral Equation method. In 3D, the fluid velocity at a target point is given by a 2D surface integral over all interfaces, thus reducing the dimension of the problem. A core challenge is that for target points near, but not on, an interface, the surface integral is near-singular and standard quadratures lose accuracy. This paper presents a method to accurately compute the near-singular integrals arising in elliptic boundary value problems in 3D. It is based on a local series approximation of the integrand about a base point on the surface, obtained by orthogonal projection of the target point onto the surface. The elementary functions in the resulting series approximation can be integrated to high accuracy in a neighborhood of the base point using a recursive algorithm. The remaining integral is evaluated numerically using a standard quadrature rule, chosen here to be the 4th order Trapezoidal rule. The method is reduced to the standard quadrature plus a correction, and is uniformly of 4th order. The method is applied to resolve Stokes flow past several ellipsoidal rigid bodies. We compare the error in the velocity near the bodies, and in the time and displacement of particles traveling around the bodies, computed with and without the corrections.

math.NA↗
arXiv2026

A coupled HDG discretization for the interaction between acoustic and elastic waves

We propose and analyze an HDG scheme for the Laplace-domain interaction between a transient acoustic wave and a bounded elastic solid embedded in an unbounded fluid medium. The elastic and acoustic domains are coupled through transmission conditions derived from the continuity of the normal stress and of the normal component of the velocities at the interface. The analysis of the HDG discretization of the coupled weak formulation is the main focus of the article. Two mixed variables (the stress tensor and the velocity of the acoustic wave) are included, while the symmetry of the stress tensor is imposed weakly by considering the antisymmetric part of the strain tensor (the spin or vorticity tensor) as an additional unknown. Convergence of the method is demonstrated and theoretical rates are obtained; numerical results suggesting optimal order of convergence and superconvergence of the traces are presented.

math.NA↗