arXiv subjects

Peihao Zhang

Publications and source records attributed to Peihao Zhang.

7 recordsLinked to original sources

arXiv2026

High-order finite element method for perfect conductivity and linear elasticity with nearly touching inclusions

In perfect conductivity and linear elasticity problems, the electric field and stress always become highly concentrated within narrow regions between adjacent perfect (rigid) inclusions, and blow up as the distance between inclusions approaches zero. The design of high-order numerical methods with rigorous error analysis for such concentration problems remains open. In this paper, we present the first high-order finite element method for solving these problems. Our approach is based on asymptotic estimates of high-order derivatives of solutions, employing a graded mesh and auxiliary basis functions specifically designed from these derivative estimates. We prove that the proposed method converges with an $H^1$-error bound of $O(h^p)$ and the error bound is independent of the distance (possibly approaching zero) between inclusions, where $h$ is the mesh size and $p$ is the degree of finite elements. Numerical examples in both two and three dimensions are presented to demonstrate the convergence rates of the numerical solutions. In particular, the blow-up behaviors of the gradients of solutions are demonstrated when the inclusions approach each other.

math.NA↗

arXiv2025

Higher derivative estimates for Stokes equations with closely spaced rigid inclusions in three dimensions

In this paper, we establish higher-order derivative estimates for the Stokes equations in a three-dimensional domain containing two closely spaced rigid inclusions. We construct a sequence of auxiliary functions via an inductive process to isolate the leading singular terms of higher-order derivatives within the narrow region between the inclusions. For a class of convex inclusions of general shapes, the construction of three-dimensional auxiliary functions -- unlike the two-dimensional case -- relies on the decay properties of solutions to a class of two-dimensional partial differential equations with singular coefficients. Taking advantage of this, we obtain pointwise upper bounds of derivatives up to the seventh order for general inclusions. Under additional symmetry conditions, we derive optimal estimates for derivatives of arbitrary order. Consequently, we obtain precise blow-up rates for the Cauchy stress and its higher-order derivatives in the narrow region between the inclusions.

math.AP↗

arXiv2024

Optimal higher derivative estimates for Stokes equations with closely spaced rigid inclusions

In this paper, we study the interaction between two closely spaced rigid inclusions suspended in a Stokes flow. It is well known that the stress significantly amplifies in the narrow region between the inclusions as the distance between them approaches zero. To gain deeper insight into these interactions, we derive high-order derivative estimates for the Stokes equation in the presence of two rigid inclusions in two dimensions. Our approach resonates with the method used to handle the incompressibility constraint in the standard convex integration scheme. Under certain symmetric assumptions on the domain, these estimates are shown to be optimal. As a result, we establish the precise blow-up rates of the Cauchy stress and its higher-order derivatives in the narrow region.

math.AP↗

arXiv2024

Stress concentration between two adjacent rigid particles in Navier-Stokes flow

In this paper we investigate the stress concentration problem that occurs when two convex rigid particles are closely immersed in a fluid flow. The governing equations for the fluid flow are the stationary incompressible Navier-Stokes equations. We establish precise upper bounds for the gradients and second-order derivatives of the fluid velocity as the distance between particles approaches zero, in dimensions two and three. The optimality of these blow-up rates of the gradients is demonstrated by deriving corresponding lower bounds. New difficulties arising from the nonlinear term in the Navier-Stokes equations is overcome. Consequently, the blow up rates of the Cauchy stress are studied as well.

math.AP↗

arXiv2024

Optimal higher derivative estimates for solutions of the Lamé system with closely spaced hard inclusions

We investigate higher derivative estimates for the Lamé system with hard inclusions embedded in a bounded domain in $\mathbb{R}^{d}$. As the distance $\varepsilon$ between two closely spaced hard inclusions approaches zero, the stress in the narrow regions between the inclusions increases significantly. This stress is captured by the gradient of the solution. The key contribution of this paper is a detailed characterization of this singularity, achieved by deriving higher derivative estimates for solutions to the Lamé system with partially infinite coefficients. These upper bounds are shown to be sharp in two and three dimensions when the domain exhibits certain symmetries. To the best of our knowledge, this is the first work to precisely quantify the singular behavior of higher derivatives in the Lamé system with hard inclusions.

math.AP↗

arXiv2023

Stress blow-up analysis when suspending rigid particles approach boundary in 3D Stokes flow

The stress concentration is a common phenomenon in the study of fluid-solid model. In this paper, we investigate the boundary gradient estimates and the second order derivatives estimates for the Stokes flow when the rigid particles approach the boundary of the matrix in dimension three. We classify the effect on the blow-up rates of the stress from the prescribed various boundary data: locally constant case and locally polynomial case. Our results hold for general convex inclusions, including two important cases in practice, spherical inclusions and ellipsoidal inclusions. The blow-up rates of the Cauchy stress in the narrow region are also obtained. We establish the corresponding estimates in higher dimensions greater than three.

math.AP↗

arXiv2023

Stress blow-up analysis when a suspending rigid particle approaches the boundary in Stokes flow: 2D case

It is an interesting and important topic to study the motion of small particles in a viscous liquid in current applied research. In this paper we assume the particles are convex with arbitrary shapes and mainly investigate the interaction between the rigid particles and the domain boundary when the distance tends to zero. In fact, even though the domain and the prescribed boundary data are both smooth, it is possible to cause a definite increase of the blow-up rate of the stress. This problem has the free boundary value feature due to the rigidity assumption on the particle. We find that the prescribed local boundary data directly affects on the free boundary value on the particle. Two kinds of boundary data are considered: locally constant boundary data and locally polynomial boundary data. For the former we prove the free boundary value is close to the prescribed constant, while for the latter we show the influence on the blow-up rate from the order of growth of the prescribed polynomial. Based on pointwise upper bounds in the neck region and lower bounds at the midpoint of the shortest line between the particle and the domain boundary, we show that these blow-up rates obtained in this paper are optimal. These precise estimates will help us understand the underlying mechanism of the hydrodynamic interactions in fluid particle model.

math.AP↗