arXiv subjects

Qilong Zhai

Publications and source records attributed to Qilong Zhai.

At least 19 recordsLinked to original sources

arXiv2026

Eigenvalue Asymptotics in High-Contrast Media

We investigate the eigenfrequency asymptotics of a bounded acoustic cavity containing a shrinking high-contrast inclusion. This configuration is motivated by contrast-enhanced ultrasound imaging, in which microbubbles are employed as acoustic contrast agents. We identify dimension-dependent material scalings that keep the inclusion-induced resonance in a fixed order-one frequency regime as the inclusion shrinks. Our main result gives a complete asymptotic description of the spectrum in any prescribed bounded frequency window. Two distinct spectral mechanisms arise: the background Dirichlet eigenfrequencies persist as perturbed eigenvalue clusters, while the singular material contrast creates an additional Minnaert eigenvalue branch. The limiting Minnaert frequency is explicit in both dimensions, with a capacitance-based expression in three dimensions and an area-based expression in two dimensions. Two-dimensional numerical experiments confirm both the perturbation of the background Dirichlet eigenfrequencies and the emergence of the additional Minnaert branch.

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arXiv2026

Regularity Analysis and Tensor Neural Network Methods for Quasiperiodic Elliptic Equations

This paper investigates quasiperiodic elliptic equations using a well-established projection method, by which the original problem is reformulated as a degenerate periodic variational problem defined on a higher-dimensional torus. The main analytical difficulty lies in the degeneracy of the projected torus problem: the variational formulation is coercive in the projected Sobolev spaces induced by the projection matrix, but fails to be coercive with respect to the standard Sobolev norm. We establish the well-posedness and regularity estimates within these projected Sobolev spaces. However, such projected Sobolev regularity alone is insufficient for standard spectral approximation: we show that it may result in arbitrarily slow Fourier convergence. To overcome this limitation, under a Diophantine condition for the projection matrix, together with appropriate assumptions on the coefficients and source term, we derive improved Sobolev regularity of the solution in standard Sobolev spaces.This provides a systematic mechanism for justifying the Sobolev regularity required in Fourier spectral convergence analysis, rather than imposing it a priori. Consequently, the improved Sobolev regularity enables quantitative Fourier projection error estimates for quasiperiodic problems. Inspired by these analytical results, we propose an adaptive tensor neural network Galerkin method that is naturally tailored to this degenerate high-dimensional periodic problem. Thanks to its tensor-product structure, high-dimensional variational integrals can be fully decoupled into one-dimensional quadratures, avoiding Monte Carlo sampling and yielding accurate deterministic numerical results. Numerical experiments on several quasiperiodic elliptic problems demonstrate the accuracy and efficiency of the proposed method.

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arXiv2026

Quantum Simulation of Stokes Flow via Schrödingerisation and Artificial Compressibility

Simulating incompressible Stokes flow is essential for studies in microfluidics and low-Reynolds number hydrodynamics. However, the computational cost of resolving the associated saddle-point problem grows prohibitively with the dimensionality of the problem. In this work, we present a quantum algorithm based on the Schrödingerisation technique for the Stokes equations, incorporating an artificial compressibility regularization. The core of our approach is the design of an explicit quantum circuit that encodes the resulting regularized system. The artificial compressibility formulation provides a unified framework for the system, which is then efficiently mapped to a quantum circuit via the Schrödingerisation procedure. A rigorous complexity analysis demonstrates the quantum computational advantage of our algorithms in high-dimensional settings, notably an exponential speedup in problem dimensionality. The validity and scalability of the proposed method are corroborated by numerical simulations performed on Qiskit.

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arXiv2026

The Immersed Discontinuous Galerkin Method for Elliptic Interface Problems

This paper is devoted to construction and convergence analysis of the linear explicit immersed finite element (IFE) function. For the interface elements, the proposed IFE functions precisely satisfy the interface conditions on the actual interface. The IFE functions are constructed in an explicit form and can be obtained directly without solving any auxiliary problems or local linear systems. Although the constructed IFE functions are non-polynomial, we establish rigorous theoretical analysis showing that they achieve optimal approximation properties and satisfy the essential trace inequalities. And the constants in the analysis are independent of how the interface cuts through the elements. Based on these IFE functions, an immersed discontinuous Galerkin numerical scheme is developed. Several numerical experiments are implemented to confirm that both the IFE functions and the numerical method achieve optimal convergence rates in the $H^1$ and $L^2$ norms. Furthermore, the numerical results indicate that the condition numbers of the stiffness matrices are robust with respect to the interface location.

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arXiv2026

The weak Galerkin method for a class of Gross-Pitaevskii type eigenvalue problems

This paper aims to employ the weak Galerkin method to solve a class of nonlinear eigenvalue problems. We proved the weak Galerkin scheme produces lower bound for the energy. Moreover, by the post-processing technique, we obtain lower bound for the ground state eigenvalue. Finally, numerical experiments are provided to validate the theoretical analysis.

math.NA↗

arXiv2026

Quasiperiodic Elliptic Operators: Projection Method and Convergence Analysis

Quasiperiodic elliptic operators (QEOs) serve as fundamental models in both mathematics and physics, as exemplified by their role in the numerical modeling of one-dimensional photonic quasicrystals. However, distinct from periodic elliptic operators, approximating eigenpairs for QEOs poses significant challenges, particularly in capturing the full spectral structure (notably the continuous spectrum) and deriving convergence guarantees in the absence of compactness. In this paper, we develop a high-accuracy numerical method to compute eigenpairs of QEOs based on the projection method, which embeds quasiperiodic operators into a higher-dimensional periodic torus. To address the non-compactness issue, we construct a directional-derivative Hilbert space along irrational manifolds of a high-dimensional torus and characterize operators equivalent to QEOs within this space. By integrating spectral theory for non-compact operators into the Babuška-Osborn eigenproblem framework, we establish rigorous convergence analysis and prove that our method achieves spectral accuracy. Numerical experiments validate the accuracy and efficiency of the proposed method, including a one-dimensional photonic quasicrystal and two- and three-dimensional QEOs.

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arXiv2025

An energy- and helicity-conserving enriched galerkin method for the incompressible Navier-Stokes equations

We develop an enriched Galerkin (EG) method for the incompressible Navier-Stokes equations that conserves both kinetic energy and helicity in the inviscid limit without introducing any additional projection variables. The method employs an EG velocity space, which is the first-order continuous Galerkin space enriched with piecewise constants defined on mesh faces, together with piecewise-constant pressure. Two numerical schemes based on the rotational form of the convective term are proposed: a nonlinear scheme and a linear variant. Both schemes exactly preserve the discrete helicity and kinetic energy, and the Picard iteration maintains the conservation properties of the nonlinear scheme. We prove the conservation properties of both the methods, and establish stability and rigorous error estimates for the nonlinear scheme. Numerical examples demonstrate the accuracy and conservation of the proposed linear scheme.

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arXiv2025

A posteriori error estimation for weak Galerkin method of the fourth-order singularly perturbed problem

In this paper, we present a posteriori error estimation for weak Galerkin method applied to fourth order singularly perturbed problem. The weak Galerkin discretization space and numerical scheme are first described. A fully computable residual type error estimator is then constructed. Both the reliability and efficiency of the proposed estimator are rigorously demonstrated. Numerical experiments are provided to validate the theoretical findings.

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arXiv2025

The Immersed Skeletal Finite Element Method for Elliptic Interface Problems

In this paper, we present a new immersed finite element scheme for solving elliptic interface problems on unfitted meshes by combining the skeletal finite element method (FEM) with the standard FEM. The skeletal FEM is used for the interface elements. In other words, we take piecewise functions as the unknowns inside the interface element and on its boundary. We employ the immersed finite element functions as interior functions that precisely satisfy the interface conditions. On the interface edges, we define two boundary functions to capture the discontinuity. The Lagrange element is used for the non-interface elements. The proposed scheme is simple and flexible. We prove that this scheme achieves optimal convergence orders in both the $H^1$ norm and $L^2$ norm. Numerical experiments are presented to demonstrate the efficiency and accuracy of the proposed method.

math.NA↗

arXiv2025

The Weak Galerkin and Crouzeix-Raviart element method for elastic eigenvalue problems

In this paper, we first introduce an abstract framework to solve the eigenvalue problem by weak Galerkin (WG) method. By the application of the framework, WG method is proved to be locking-free and gives asymptotic lower bounds for the elastic eigenvalue problem. Also, we analyze the lower bound property for Crouzeix-Raviart (CR) element as an extensional work. In the end, we present some numerical experiments to support the theoretical results.

math.NA↗

arXiv2025

The two-grid weak Galerkin method and enriched Crouzeix-Raviart element method for linear elastic eigenvalue problems

In this paper, we present a two-gird skill to accelerate the weak Galerkin method. By the proper use of parameters, the two-grid weak Galerkin method not only doubles the convergence rate, but also maintains the asymptotic lower bounds property of the weak Galerkin (WG) method. Moreover, we propose an enriched Crouzeix-Raviart (ECR) scheme, which can also provide lower bounds for the linear elastic eigenvalue problems.

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arXiv2024

The pressure-robust weak Galerkin finite element method for Stokes-Darcy problem

In this paper, we propose a pressure-robust weak Galerkin (WG) finite element scheme to solve the Stokes-Darcy problem. To construct the pressure-robust numerical scheme, we use the divergence-free velocity reconstruction operator to modify the test function on the right side of the numerical scheme. We prove the error between the velocity function and its numerical solution is independent of the pressure function and viscosity coefficient. Moreover, the errors of the velocity function and the pressure function reach the optimal convergence orders under the energy norm, as validated by both theoretical analysis and numerical results.

math.NA↗

arXiv2024

Convergence analysis of a weak Galerkin finite element method on a Bakhvalov-type mesh for a singularly perturbed convection-diffusion equation in 2D

In this paper, we propose a weak Galerkin finite element method (WG) for solving singularly perturbed convection-diffusion problems on a Bakhvalov-type mesh in 2D. Our method is flexible and allows the use of discontinuous approximation functions on the meshe. An error estimate is devised in a suitable norm and the optimal convergence order is obtained. Finally, numerical experiments are given to support the theory and to show the efficiency of the proposed method.

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arXiv2024

A weak Galerkin finite element method for solving the asymptotic lower bound of Maxwell eigenvalue problem

In this paper, we propose a weak Galerkin (WG) finite element method for the Maxwell eigenvalue problem. By restricting subspaces, we transform the mixed form of Maxwell eigenvalue problem into simple elliptic equation. Then we give the WG numerical scheme for the Maxwell eigenvalue problem. Furthermore, we obtain the optimal error estimates of arbitrarily high convergence order and prove the lower bound property of numerical solutions for eigenvalues. Numerical experiments show the accuracy of theoretical analysis and the property of lower bound.

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arXiv2024

The weak Galerkin finite element method for the Steklov eigenvalue problem

This paper introduces the application of the weak Galerkin (WG) finite element method to solve the Steklov eigenvalue problem, focusing on obtaining lower bounds of the eigenvalues. The noncomforming finite element space of the weak Galerkin finite element method is the key to obtain lower bounds of the eigenvalues. The arbitary high order lower bound estimates are given and the guaranteed lower bounds of the eigenvalues are also discussed. Numerical results demonstrate the accuracy and lower bound property of the numerical scheme.

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arXiv2024

The weak Galerkin finite element method for Stokes interface problems with curved interface

In this paper, we develop a new weak Galerkin finite element scheme for the Stokes interface problem with curved interfaces. We take a unique vector-valued function at the interface and reflect the interface condition in the variational problem. Theoretical analysis and numerical experiments show that the errors can reach the optimal convergence order under the energy norm and $L^2$ norm.

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arXiv2024

Convergence analysis of a weak Galerkin finite element method on a Shishkin mesh for a singularly perturbed fourth-order problem in 2D

We consider the singularly perturbed fourth-order boundary value problem $\varepsilon ^{2}Δ^{2}u-Δu=f $ on the unit square $Ω\subset \mathbb{R}^2$, with boundary conditions $u = \partial u / \partial n = 0$ on $\partial Ω$, where $\varepsilon \in (0, 1)$ is a small parameter. The problem is solved numerically by means of a weak Galerkin(WG) finite element method, which is highly robust and flexible in the element construction by using discontinuous piecewise polynomials on finite element partitions consisting of polygons of arbitrary shape. The resulting WG finite element formulation is symmetric, positive definite, and parameter-free. Under reasonable assumptions on the structure of the boundary layers that appear in the solution, a family of suitable Shishkin meshes with $N^2$ elements is constructed ,convergence of the method is proved in a discrete $H^2$ norm for the corresponding WG finite element solutions and numerical results are presented.

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arXiv2024

A stabilizer free weak Galerkin method with implicit $θ$-schemes for fourth order parabolic problems

In this paper, we combine the stabilizer free weak Galerkin (SFWG) method and the implicit $θ$-schemes in time for $θ\in [\frac{1}{2},1]$ to solve the fourth-order parabolic problem. In particular, when $θ=1$, the full-discrete scheme is first-order backward Euler and the scheme is second-order Crank Nicolson scheme if $θ=\frac{1}{2}$. Next, we analyze the well-posedness of the schemes and deduce the optimal convergence orders of the error in the $H^2$ and $L^2$ norms. Finally, numerical examples confirm the theoretical results.

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