arXiv subjects

Shuonan Wu

Publications and source records attributed to Shuonan Wu.

At least 19 recordsLinked to original sources

arXiv2026

Stabilized Morley FEM for surface Stokes in stream-function formulation: Optimal convergence via a new geometric estimate

We propose an intrinsic stabilized Morley finite element method for the stream-function formulation of the surface Stokes problem on closed surfaces. The method is posed directly on a polyhedral approximation of the surface. A parameter-free jump stabilization is introduced to recover coercivity, and a discrete Korn's inequality for the trace-free Hessian is established. The main analytical contribution is a normal-separated geometric estimate for piecewise linearly approximated closed surfaces. It shows that a broad class of normal-dependent geometric consistency errors is in fact second order, even though a direct treatment suggests only first-order control. The missing order is recovered through an integral cancellation in the first normal variation. This estimate refines the standard $P_hν$-type estimate and yields second-order consistency for the trace-free Hessian, the Stokes-type tensor Green identity, and the conormal fluxes arising in the discretization. Together with the discrete Korn's inequality, these estimates yield, under the natural $H^3$ regularity, optimal-order convergence: first order in the broken $H^2$ norm and second order in the broken $H^1$ norm. As a direct consequence, the recovered tangential velocity admits a second-order $L^2$ estimate. Numerical experiments are provided to support the theoretical results.

math.NA↗

arXiv2025

Hierarchical and ultrametric barriers in the energy landscape of jammed granular matter

According to the mean-field glass theory, the (free) energy landscape of disordered systems is hierarchical and ultrametric if they belong to the full-replica-symmetry-breaking universality class. However, examining this theoretical picture in three-dimensional systems remains challenging, where the energy barriers become finite. Here, we numerically explore the energy landscape of granular models near the jamming transition using a saddle dynamics algorithm to locate both local energy minima and saddles. The multi-scale distances and energy barriers between minima are characterized by two metrics, both of which exhibit signatures of an ultrametric space. The scale-free distribution of energy barriers reveals that the landscape is hierarchical.

cond-mat.soft↗

arXiv2025

Local discontinuous Galerkin method for the integral fractional Laplacian

We develop and analyze a local discontinuous Galerkin (LDG) method for solving integral fractional Laplacian problems on bounded Lipschitz domains. The method is based on a three-field mixed formulation involving the primal variable, its gradient, and the corresponding Riesz potential, yielding a flux-based structure well suited for LDG discretizations while retaining the intrinsic nonlocal interaction. A key ingredient of our analysis is a rigorous study of the weighted Hölder and Sobolev regularity of the Riesz potential, which enables accurate characterization of boundary singularities. Guided by these regularity results, we propose LDG schemes on quasi-uniform and graded meshes, with additional stabilization in the graded case to reconcile the discrepancy between the discrete spaces for the Riesz potential and flux fields. Optimal a priori error estimates are established, and numerical experiments corroborate the theoretical results.

math.NA↗

arXiv2025

A streamline upwind/Petrov-Galerkin method for the magnetic advection-diffusion problem

This paper presents the development and analysis of a streamline upwind/Petrov-Galerkin (SUPG) method for the magnetic advection-diffusion problem. A key feature of the method is an SUPG-type stabilization term based on the residuals and weighted advection terms of the test function. By introducing a lifting operator to characterize the jumps of finite element functions across element interfaces, we define a discrete magnetic advection operator, which subsequently enables the formulation of the desired SUPG method. Under mild assumptions, we establish the stability of the scheme and derive optimal error estimates. Numerical examples in both two and three dimensions are provided to demonstrate the theoretical convergence and stabilization properties of the proposed method.

math.NA↗

arXiv2025

Local projection stabilization methods for $\boldsymbol{H}({\rm curl})$ and $\boldsymbol{H}({\rm div})$ advection problems

We devise local projection stabilization (LPS) methods for advection problems in the $\boldsymbol{H}$(curl) and $\boldsymbol{H}$(div) spaces, employing conforming finite element spaces of arbitrary order within a unified framework. The key ingredient is a local inf-sup condition, enabled by enriching the approximation space with appropriate $\boldsymbol{H}$(d) bubble functions (with d = curl or div). This enrichment allows for the construction of modified interpolation operators, which are crucial for establishing optimal a priori error estimates in the energy norm. Numerical examples are presented to verify both the theoretical results and the stabilization properties of the proposed method.

math.NA↗

arXiv2024

$\mathcal{P}_m$ Interior Penalty Nonconforming Finite Element Methods for $2m$-th Order PDEs in $\mathbb{R}^n$

In general $n$-dimensional simplicial meshes, we propose a family of interior penalty nonconforming finite element methods for $2m$-th order partial differential equations, where $m \geq 0$ and $n \geq 1$. For this family of nonconforming finite elements, the shape function space consists of polynomials with a degree not greater than $m$, making it minimal. This family of finite element spaces exhibits natural inclusion properties, analogous to those in the corresponding Sobolev spaces in the continuous case. By applying interior penalty to the bilinear form, we establish quasi-optimal error estimates in the energy norm. Due to the weak continuity of the nonconforming finite element spaces, the interior penalty terms in the bilinear form take a simple form, and an interesting property is that the penalty parameter needs only to be a positive constant of $\mathcal{O}(1)$. These theoretical results are further validated by numerical tests.

math.NA↗

arXiv2024

A stabilized nonconforming finite element method for the surface biharmonic problem

This paper presents a novel stabilized nonconforming finite element method for solving the surface biharmonic problem. The method extends the New-Zienkiewicz-type (NZT) element to polyhedral (approximated) surfaces by employing the Piola transform to establish the connection of vertex gradients across adjacent elements. Key features of the surface NZT finite element space include its $H^1$-relative conformity and weak $H({\rm div})$ conformity, allowing for stabilization without the use of artificial parameters. Under the assumption that the exact solution and the dual problem possess only $H^3$ regularity, we establish optimal error estimates in the energy norm and provide, for the first time, a comprehensive analysis yielding optimal second-order convergence in the broken $H^1$ norm. Numerical experiments are provided to support the theoretical results.

math.NA↗

arXiv2024

A construction of canonical nonconforming finite element spaces for elliptic equations of any order in any dimension

A unified construction of canonical $H^m$-nonconforming finite elements is developed for $n$-dimensional simplices for any $m, n \geq 1$. Consistency with the Morley-Wang-Xu elements [Math. Comp. 82 (2013), pp. 25-43] is maintained when $m \leq n$. In the general case, the degrees of freedom and the shape function space exhibit well-matched multi-layer structures that ensure their alignment. Building on the concept of the nonconforming bubble function, the unisolvence is established using an equivalent integral-type representation of the shape function space and by applying induction on $m$. The corresponding nonconforming finite element method applies to $2m$-th order elliptic problems, with numerical results for $m=3$ and $m=4$ in 2D supporting the theoretical analysis.

math.NA↗

arXiv2024

Solution landscape of reaction-diffusion systems reveals a nonlinear mechanism and spatial robustness of pattern formation

Spontaneous pattern formation in homogeneous systems is ubiquitous in nature. Although Turing demonstrated that spatial patterns can emerge in reaction-diffusion (RD) systems when the homogeneous state becomes linearly unstable, it remains unclear whether the Turing mechanism is the only route for pattern formation. Here, we develop an efficient algorithm to systematically map the solution landscape to find all steady-state solutions. By applying our method to generic RD models, we find that stable spatial patterns can emerge via saddle-node bifurcations before the onset of Turing instability. Furthermore, by using a generalized action in functional space based on large deviation theory, our method is extended to evaluate stability of spatial patterns against noise. Applying this general approach in a three-species RD model, we show that though formation of Turing patterns only requires two chemical species, the third species is critical for stabilizing patterns against strong intrinsic noise in small biochemical systems.

physics.bio-ph↗

arXiv2024

A robust solver for H(curl) convection-diffusion and its local Fourier analysis

In this paper, we present a robust and efficient multigrid solver based on an exponential-fitting discretization for 2D H(curl) convection-diffusion problems. By leveraging an exponential identity, we characterize the kernel of H(curl) convection-diffusion problems and design a suitable hybrid smoother. This smoother incorporates a lexicographic Gauss-Seidel smoother within a downwind type and smoothing over an auxiliary problem, corresponding to H(grad) convection-diffusion problems for kernel correction. We analyze the convergence properties of the smoothers and the two-level method using local Fourier analysis (LFA). The performance of the algorithms demonstrates robustness in both convection-dominated and diffusion-dominated cases.

math.NA↗

arXiv2023

Exponentially-fitted finite elements for $H({\rm curl})$ and $H({\rm div})$ convection-diffusion problems

This paper presents a novel approach to the construction of the lowest order $H(\mathrm{curl})$ and $H(\mathrm{div})$ exponentially-fitted finite element spaces ${\mathcal{S}_{1^-}^{k}}~(k=1,2)$ on 3D simplicial mesh for corresponding convection-diffusion problems. It is noteworthy that this method not only facilitates the construction of the functions themselves but also provides corresponding discrete fluxes simultaneously. Utilizing this approach, we successfully establish a discrete convection-diffusion complex and employ a specialized weighted interpolation to establish a bridge between the continuous complex and the discrete complex, resulting in a coherent framework. Furthermore, we demonstrate the commutativity of the framework when the convection field is locally constant, along with the exactness of the discrete convection-diffusion complex. Consequently, these types of spaces can be directly employed to devise the corresponding discrete scheme through a Petrov-Galerkin method.

math.NA↗

arXiv2023

A Monotone Discretization for the Fractional Obstacle Problem

We introduce a novel monotone discretization method for addressing obstacle problems involving the integral fractional Laplacian with homogeneous Dirichlet boundary conditions over bounded Lipschitz domains. This problem is prevalent in mathematical finance, particle systems, and elastic theory. By leveraging insights from the successful monotone discretization of the fractional Laplacian, we establish uniform boundedness, solution existence, and uniqueness for the numerical solutions of the fractional obstacle problem. We employ a policy iteration approach for efficient solution of discrete nonlinear problems and prove its finite convergence. Our improved policy iteration, adapted to solution regularity, demonstrates superior performance by modifying discretization across different regions. Numerical examples underscore the method's efficacy.

math.NA↗

arXiv2023

A hybridizable discontinuous Galerkin method for magnetic advection-diffusion problems

We propose and analyze a hybridizable discontinuous Galerkin (HDG) method for solving a mixed magnetic advection-diffusion problem within a more general Friedrichs system framework. With carefully constructed numerical traces, we introduce two distinct stabilization parameters: $τ_t$ for the tangential trace and $τ_n$ for the normal trace. These parameters are tailored to satisfy different requirements, ensuring the stability and convergence of the method. Furthermore, we incorporate a weight function to facilitate the establishment of stability conditions. We also investigate an elementwise postprocessing technique that proves to be effective for both two-dimensional and three-dimensional problems in terms of broken $H({\rm curl})$ semi-norm accuracy improvement. Extensive numerical examples are presented to showcase the performance and effectiveness of the HDG method and the postprocessing techniques.

math.NA↗

arXiv2023

Two families of $n$-rectangle nonconforming finite elements for sixth-order elliptic equations

In this paper, we propose two families of nonconforming finite elements on $n$-rectangle meshes of any dimension to solve the sixth-order elliptic equations. The unisolvent property and the approximation ability of the new finite element spaces are established. A new mechanism, called the exchange of sub-rectangles, for investigating the weak continuities of the proposed elements is discovered. With the help of some conforming relatives for the $H^3$ problems, we establish the quasi-optimal error estimate for the tri-harmonic equation in the broken $H^3$ norm of any dimension. The theoretical results are validated further by the numerical tests in both 2D and 3D situations.

math.NA↗

arXiv2022

A monotone discretization for integral fractional Laplacian on bounded Lipschitz domains: Pointwise error estimates under Hölder regularity

We propose a monotone discretization for the integral fractional Laplace equation on bounded Lipschitz domains with the homogeneous Dirichlet boundary condition. The method is inspired by a quadrature-based finite difference method of Huang and Oberman, but is defined on unstructured grids in arbitrary dimensions with a more flexible domain for approximating singular integral. The scale of the singular integral domain not only depends on the local grid size, but also on the distance to the boundary, since the Hölder coefficient of the solution deteriorates as it approaches the boundary. By using a discrete barrier function that also reflects the distance to the boundary, we show optimal pointwise convergence rates in terms of the Hölder regularity of the data on both quasi-uniform and graded grids. Several numerical examples are provided to illustrate the sharpness of the theoretical results.

math.NA↗

arXiv2022

Discontinuous Galerkin methods for magnetic advection-diffusion problems

We devise and analyze a class of the primal discontinuous Galerkin methods for the magnetic advection-diffusion problems based on the weighted-residual approach. In addition to the upwind stabilization, we find a new mechanism under the vector case that provides more flexibility in constructing the schemes. For the more general Friedrichs system, we show the stability and optimal error estimate, which boil down to two core ingredients -- the weight function and the special projection -- that contain information of advection. Numerical experiments are provided to verify the theoretical results.

math.NA↗

arXiv2022

A $C^{0}$ finite element approximation of planar oblique derivative problems in non-divergence form

This paper proposes a $C^{0}$ (non-Lagrange) primal finite element approximation of the linear elliptic equations in non-divergence form with oblique boundary conditions in planar, curved domains. As an extension of [Calcolo, 58 (2022), No. 9], the Miranda-Talenti estimate for oblique boundary conditions at a discrete level is established by enhancing the regularity on the vertices. Consequently, the coercivity constant for the proposed scheme is exactly the same as that from PDE theory. The quasi-optimal order error estimates are established by carefully studying the approximation property of the finite element spaces. Numerical experiments are provided to verify the convergence theory and to demonstrate the accuracy and efficiency of the proposed methods.

math.NA↗

arXiv2021

Auxiliary Space Preconditioners for $C^{0}$ Finite Element Approximation of Hamilton--Jacobi--Bellman Equations with Cordes Coefficients

In the past decade, there are many works on the finite element methods for the fully nonlinear Hamilton--Jacobi--Bellman (HJB) equations with Cordes condition. The linearised systems have large condition numbers, which depend not only on the mesh size, but also on the parameters in the Cordes condition. This paper is concerned with the design and analysis of auxiliary space preconditioners for the linearised systems of $C^0$ finite element discretization of HJB equations [Calcolo, 58, 2021]. Based on the stable decomposition on the auxiliary spaces, we propose both the additive and multiplicative preconditoners which converge uniformly in the sense that the resulting condition number is independent of both the number of degrees of freedom and the parameter $λ$ in Cordes condition. Numerical experiments are carried out to illustrate the efficiency of the proposed preconditioners.

math.NA↗