arXiv subjects

Buyang Li

Publications and source records attributed to Buyang Li.

At least 19 recordsLinked to original sources

arXiv2026

Convergence of a CutFEM for fluid--structure interaction with a deforming interface

We present a rigorous error analysis for a semidiscrete immersed CutFEM applied to fluid--structure interaction (FSI) problems involving an incompressible viscous fluid and an elastic solid. The method employs a Lagrange multiplier to enforce the fluid-solid coupling conditions while fully accounting for the effect of the structural deformation on the fluid domain. For sufficiently regular FSI solutions and finite elements of degree $k\geq3$, we prove convergence of order $k$ for the fluid velocity and solid displacement and convergence of order $k-1$ for the fluid pressure in the energy norms associated with the FSI problem. To the best of our knowledge, this is the first rigorous convergence analysis of an unfitted finite element method for a fully coupled FSI system with a deforming interface.

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arXiv2026

An $H^{-1}$ least-squares UnCut FEM on domains defined by a level set function

We propose a novel UnCut finite element method (FEM) for the Poisson and Stokes equations on domains with curved boundaries represented by a level set function. Like the $\phi$-FEM, the method avoids numerical integration over cut subregions of boundary elements, but introduces a novel least-squares formulation that minimizes an $H^{-1}$ residual of the governing equations. This formulation ensures stability without requiring large stabilization parameters, thereby eliminating the need for user-tuned penalty parameters and improving the robustness of the computation. Optimal-order convergence of the UnCut FEM solutions is rigorously established in the $H^1$ norm for both the Poisson and Stokes equations, and numerical experiments are presented to support the theoretical analysis.

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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.

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arXiv2026

Weighted $H^2$ regularity of fluid-structure interaction when the interface intersects the boundary

This paper analyzes the regularity of solutions to a fluid-structure interaction (FSI) problem involving a Stokesian fluid and a linear elastic solid in a two-dimensional polygonal domain, seperated by an interface which intersects the boundary of the domain. The main challenges arise from the limited solution regularity caused by geometric singularities at the domain corners. To address this, we introduce tailored weighted Sobolev spaces, denoted by $\mathbf{H}^{k,l}_{\boldsymbol{\beta}}$, and a novel solution operator framework on these spaces. This framework provides the key insight for decoupling the coupled FSI system into tractable fluid and elastic solid subproblems in the regularity analysis. As our main result, we prove the existence and uniqueness of a solution in the weighted Sobolev space $\mathbf{H}^{2,2}_{\boldsymbol{\beta}}$.

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arXiv2026

Convergence of Finite Element Methods for Ricci Flow

The convergence of a finite element discretization for the two-dimensional Ricci flow is proved. In this method, the Ricci flow on a two-dimensional surface is formulated into solution-driven metric evolution, with the metric evolution driven by the Gauss curvature. The Gauss curvature satisfies a parabolic equation that in turn depends on the metric, thereby enhancing the parabolic structure of the problem. The solution-driven metric evolution formulation is discretized by the finite element method, and the convergence of finite element approximations is proved by adapting the matrix-vector formulation developed in the literature initially for studying solution-driven surface evolution in extrinsic curvature flow. In addition to its convergence, the proposed method also preserves important geometric structures of the Ricci flow at the discrete level, such as area conservation and the Gauss-Bonnet theorem. Extensive numerical experiments are presented to demonstrate the convergence of the proposed method as well as the simulation of Ricci flow.

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arXiv2026

Finite element exponential integration for rough solutions of nonlinear wave equations. Part I: Dirichlet boundary conditions on polygonal and polyhedral domains

We study a fully discrete scheme for nonlinear wave equations on general bounded polygonal/polyhedral domains with initial data $(u^0,v^0)\in H^\gamma(\Omega)\times H^{\gamma-1}(\Omega)$, $0<\gamma\le 1$, subject to the natural compatibility condition associated with the homogeneous Dirichlet boundary condition. The scheme combines an exponential Euler time integrator with a finite element spatial discretization. In contrast to existing low-regularity error analyses, which are mostly based on Fourier spectral discretizations, our approach applies to general bounded domains and finite element spatial discretizations. We prove rigorous error estimates for low-regularity solutions. The analysis is based on a frequency decomposition of the underlying elliptic operator, used solely as an analytical regularization device and not in the actual implementation, which allows low-regularity techniques to be extended beyond the Fourier spectral framework. The results also indicate that higher-order finite element methods remain advantageous in spatial approximation even for solutions of limited Sobolev regularity. Numerical experiments on different domains and with different polynomial degrees confirm the predicted convergence behavior.

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arXiv2026

Finite element exponential integration for rough solutions of nonlinear wave equations. Part II: Dynamic boundary conditions on curved domains

We study nonlinear wave equations with dynamic boundary conditions on smooth bounded domains and analyze a fully discrete approximation in the low-regularity regime. The method combines isoparametric bulk--surface finite elements of degree $k$ with an exponential integrator in time. Assuming only bounded energy of the exact solution, we prove convergence of the displacement--velocity pair in the weak norm $L^2(\Omega;\Gamma)\times H^{-1}(\Omega;\Gamma)$. The scheme achieves first-order convergence in time and spatial convergence of order $h^{2/3}$ for $k=1$ and $h^{(k+2)/(k+3)}$ for $k\ge 2$. In particular, these rates show that higher-order finite elements retain a provable asymptotic advantage even at low regularity. A central difficulty is that the continuous and discrete bulk--surface problems are posed on different geometries and must therefore be compared directly in weak norms. To address this, we develop a weak-norm framework for non-conforming geometries based on lift and adjoint-lift operators, combined with a frequency-decomposition argument. To the best of our knowledge, this is the first fully discrete low-regularity convergence result for nonlinear wave equations with dynamic boundary conditions in a non-conforming bulk--surface finite element setting. Numerical experiments confirm the predicted rates and illustrate the improved efficiency of higher-order methods.

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arXiv2026

Structure-preserving Lagrange-multiplier methods for mean curvature flow and their error bounds

We propose and analyze a class of evolving surface finite element methods for mean curvature flow of closed surfaces using Lagrange multipliers for preserving the energy-decreasing structure. The algorithm is based on the solution-driven formulation using Huisken's evolution equations for the normal vector and the mean curvature. The time discretizations use linearly implicit backward difference formulas (BDF) for the parabolic geometric variables and implicit Adams updates for the nodal positions. The approach also accommodates artificial tangential velocities of minimal-deformation-rate type. The resulting fully discrete algorithms are area-decreasing at every time step, with a prescribed decay rate determined by the computed mean curvature. We prove local existence and uniqueness of the discrete Lagrange multiplier and convergence of a simplified Newton iteration for its computation under weak regularity assumptions. Under stronger regularity assumptions, as used in the convergence theory for the underlying evolving surface finite element method, we derive optimal-order error bounds of order $h^k+\tau^q$ in the $H^1$-norm for finite elements of polynomial degree $k\ge 2$ and $q$-step BDF and $q$-step implicit Adams methods with $2\le q\le 5$, both without and with minimal-deformation-rate tangential motion. Numerical experiments for mean curvature flow of a sphere confirm the predicted convergence rates and show that the Lagrange-multiplier correction entails only a small computational overhead that is essentially independent of the mesh size and the time step size.

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arXiv2026

A neural network method for scalar conservation laws with convergence rates for shock-wave solutions

We propose a new entropy-compatible neural network method for scalar hyperbolic conservation laws and establish, to our knowledge, the first explicit $L^1$ convergence rates in this setting that apply to piecewise smooth entropy solutions, including those with discontinuities. The method is based on a computable approximation of the Kru\v{z}kov entropy residual that sits between the strong and weak forms of the entropy inequality. For piecewise smooth entropy solutions containing shocks, rarefactions, compound waves, regular shock interactions, and, in one space dimension, nondegenerate shock formation from smooth initial data, we construct explicit neural networks with provably small loss by combining shock-adapted continuous piecewise linear functions with known approximation properties of $\tanh$ neural networks. Together with entropy-based stability estimates, this gives rigorous $L^1$ error bounds for minimizers of the proposed loss. In particular, when the network size grows in proportion to the number of degrees of freedom of a space--time mesh of size $h$, the analysis recovers the classical Kuznetsov rate $O(h^{1/2})$ in shock-dominated cases. Numerical experiments in one and two space dimensions support the theory and suggest that the actual accuracy of the method can be better than the rate guaranteed by the analysis.

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arXiv2026

Dual formulations of geometric curvature flows and their discretizations

We propose new formulations of geometric curvature flows -- referred to as \emph{dual formulations} -- that are equivalent to the original formulations but provide a novel framework for constructing linearly implicit and energy-stable schemes for curvature-driven surface evolution, including mean curvature flow, surface diffusion, and solid-state dewetting on a substrate with a moving contact line. The dual formulations are derived by introducing, at the continuous level, an additional unknown in the form of a dual multiplier. This augmentation does not alter the continuous dynamics but makes the underlying energy-dissipation structure explicit and, in turn, enables a systematic design of linearly implicit discretizations that inherit energy stability. A key feature of this framework is that it accommodates a broad class of artificial tangential motions which can be used to maintain good mesh quality of the computed surfaces. As an illustration, we combine the framework with the minimal-deformation-rate (MDR) tangential motion, leading to what we call the \emph{dual-MDR} scheme. The resulting method is linearly implicit and energy-stable, while retaining the MDR tangential motion to maintain good mesh quality. Extensive numerical experiments demonstrate the convergence of the proposed schemes, their structure-preserving properties, and advantages on representative benchmark problems.

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arXiv2026

Finite element methods for isometric embedding of Riemannian manifolds

The isometric embedding problem for Riemannian manifolds, which connects intrinsic and extrinsic geometry, is a central question in differential geometry with deep theoretical significance and wide-ranging applications. Despite extensive analytical progress, the nonlinear and degenerate nature of this problem has hindered the development of rigorous numerical analysis in this area. As the first step toward addressing this gap, we study the numerical approximation of Weyl's problem, i.e., the isometric embedding of two-dimensional Riemannian manifolds with positive Gaussian curvature into $\mathbb{R}^3$, by establishing a new weak formulation that naturally leads to a numerical scheme well suited for high-order finite element discretization, and conducting a systematic analysis to prove the well-posedness of this weak formulation, the existence and uniqueness of its numerical solution, as well as its convergence with error estimates. This provides a foundational framework for computing isometric embeddings of Riemannian manifolds into Euclidean space, with the goal of extending it to a broader range of cases and applications in the future. Our framework also extends naturally to the isometric embedding of the Ricci flow, with rigorous error estimates, enabling the visualization of geometric evolutions in intrinsic curvature flows. Numerical experiments support the theoretical analysis by demonstrating the convergence of the method and its effectiveness in simulating isometric embeddings of given Riemannian manifolds as well as Ricci flows.

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arXiv2026

A convergent finite element method with minimal deformation rate for mean curvature flow

We propose and analyze a fully discrete parametric finite element method with minimal deformation rate (MDR) for simulating the mean curvature flow of general closed surfaces in three dimensions. The method is formulated from a coupled system that enforces the mean curvature flow law for the normal velocity while introducing an artificial tangential velocity that minimizes the deformation-rate energy, thereby preserving mesh quality without requiring remeshing or reparametrization. An $L^{2}$-projected averaged normal vector is used in the scheme to facilitate a rigorous convergence analysis. Within the projected--distance framework, we establish the first complete convergence proof for a parametric finite element method that incorporates the MDR tangential motion without relying on evolution equations for the mean curvature or the normal vector, achieving optimal-order error estimates for finite elements of degree $k \ge 3$. Numerical experiments corroborate the theoretical results and demonstrate that the proposed MDR method maintains mesh quality comparable to the Barrett--Garcke--N\"urnberg method, for which convergence has not yet been established.

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arXiv2026

An inertial minimal-deformation-rate framework for shape optimization

We propose a robust numerical framework for PDE-constrained shape optimization and Willmore-driven surface hole filling. To address two central challenges -- slow progress in flat energy landscapes, which can trigger premature stagnation at suboptimal configurations, and mesh deterioration during geometric evolution -- we couple a second-order inertial flow with a minimal-deformation-rate (MDR) mesh motion strategy. This coupling accelerates convergence while preserving mesh quality and thus avoids remeshing. To further enhance robustness for non-smooth or non-convex initial geometries, we incorporate surface-diffusion regularization within the Barrett-Garcke-N"urnberg (BGN) framework. Moreover, we extend the inertial MDR methodology to Willmore-type surface hole filling, enabling high-order smooth reconstructions even from incompatible initial data. Numerical experiments demonstrate markedly faster convergence to lower original objective values, together with consistently superior mesh preservation throughout the evolution.

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arXiv2025

Numerical analysis of 2D Navier--Stokes equations with nonsmooth initial value in the critical space

This paper addresses the numerical solution of the two-dimensional Navier--Stokes (NS) equations with nonsmooth initial data in the $L^2$ space, which is the critical space for the two-dimensional NS equations to be well-posed. In this case, the solutions of the NS equations exhibit certain singularities at $t=0$, e.g., the $H^s$ norm of the solution blows up as $t\rightarrow 0$ when $s>0$. To date, the best convergence result proved in the literature are first-order accuracy in both time and space for the semi-implicit Euler time-stepping scheme and divergence-free finite elements (even high-order finite elements are used), while numerical results demonstrate that second-order convergence in time and space may be achieved. Therefore, there is still a gap between numerical analysis and numerical computation for the NS equations with $L^2$ initial data. The primary challenge to realizing high-order convergence is the insufficient regularity in the solutions due to the rough initial condition and the nonlinearity of the equations. In this work, we propose a fully discrete numerical scheme that utilizes the Taylor--Hood or Stokes-MINI finite element method for spatial discretization and an implicit-explicit Runge--Kutta time-stepping method in conjunction with graded stepsizes. By employing discrete semigroup techniques, sharp regularity estimates, negative norm estimates and the $L^2$ projection onto the divergence-free Raviart--Thomas element space, we prove that the proposed scheme attains second-order convergence in both space and time. Numerical examples are presented to support the theoretical analysis. In particular, the convergence in space is at most second order even higher-order finite elements are used. This shows the sharpness of the convergence order proved in this article.

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arXiv2025

Computing rough solutions of the KdV equation below ${\bf L^2}$

We establish a novel numerical and analytical framework for solving the Korteweg--de Vries (KdV) equation in the negative Sobolev spaces, where classical numerical methods fail due to their reliance on high regularity and inability to control nonlinear interactions at low regularities. Numerical analysis is established by combining a continuous reformulation of the numerical scheme, the Bourgain-space estimates for the continuous reformulation, and a rescaling strategy that reduces the reformulated problem to a small initial value problem, which allow us to bridge a critical gap between numerical analysis and theoretical well-posedness by designing the first numerical method capable of solving the KdV equation in the negative Sobolev spaces. The numerical scheme is proved to have nearly optimal-order convergence with respect to the spatial degrees of freedom in the $H^{-\frac{1}{2}}$ norm for initial data in $H^s$, with $-\frac{1}{2} < s \leq 0$, a result unattainable by existing numerical methods.

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arXiv2025

An energy-stable minimal deformation rate scheme for mean curvature flow and surface diffusion

We propose a new parametric finite element method, referred to as the BGN-MDR method, for simulating both mean curvature flow and surface diffusion for closed hypersurfaces, as well as open hypersurfaces with moving contact lines in three dimensions. The method is also applicable to closed and open curves with moving contact points in two dimensions. The proposed scheme inherits the energy stability from the BGN scheme proposed by Barrett, Garcke, and N\"urnberg in 2008, and offers improved mesh quality similar to the minimal deformation rate (MDR) method proposed by Hu and Li in 2022, especially for small time step sizes where the BGN scheme may become unstable and result in deteriorated meshes.

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arXiv2025

Dynamic Ritz Projection for Finite Element Methods in Fluid-Structure Interaction

Regardless of the development of various finite element methods for fluid-structure interaction (FSI) problems, optimal-order convergence of finite element discretizations of the FSI problems in the $L^\infty(0,T;L^2)$ norm has not been proved due to the incompatibility between standard Ritz projections and the interface conditions in the FSI problems. To address this issue, we define a dynamic Ritz projection (which satisfies a dynamic interface condition) associated to the FSI problem and study its approximation properties in the $L^\infty(0,T;H^1)$ and $L^\infty(0,T;L^2)$ norms. Existence and uniqueness of the dynamic Ritz projection of the solution, as well as estimates of the error between the solution and its dynamic Ritz projection, are established. By utilizing the established results, we prove optimal-order convergence of finite element methods for the FSI problem in the $L^\infty(0,T;L^2)$ norm.

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arXiv2025

Optimal convergence of the arbitrary Lagrangian-Eulerian interface tracking method for two-phase Navier--Stokes flow without surface tension

Optimal-order convergence in the $H^1$ norm is proved for an arbitrary Lagrangian-Eulerian interface tracking finite element method for the sharp interface model of two-phase Navier-Stokes flow without surface tension, using high-order curved evolving mesh. In this method, the interfacial mesh points move with the fluid's velocity to track the sharp interface between two phases of the fluid, and the interior mesh points move according to a harmonic extension of the interface velocity. The error of the semidiscrete arbitrary Lagrangian-Eulerian interface tracking finite element method is shown to be $O(h^k)$ in the $L^\infty(0, T; H^1(\Omega))$ norm for the Taylor-Hood finite elements of degree $k \ge 2$. This high-order convergence is achieved by utilizing the piecewise smoothness of the solution on each subdomain occupied by one phase of the fluid, relying on a low global regularity on the entire moving domain. Numerical experiments illustrate and complement the theoretical results.

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