arXiv subjects

Buyang Li

Publications and source records attributed to Buyang Li.

At least 19 recordsLinked to original sources

arXiv2026

A neural network method for scalar conservation laws with convergence rates for shock waves

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 for piecewise smooth entropy solutions, including those with discontinuities. The method is based on a computable approximation of the Kružkov entropy residual that sits between the strong and weak forms of the entropy inequality. For a single regular shock in one dimension and finitely many smooth shock hypersurfaces with transversal interactions in several dimensions, 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. The one-dimensional analysis also covers rarefactions, compound waves, regular shock interactions, and nondegenerate shock formation from smooth initial data. 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 illustrate the performance of the method.

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arXiv2026

Solving the Minkowski problem by continuation from the unit sphere: spherical harmonic discretization and error estimates

Given a positive function $ρ$ on the unit sphere $\mathbb{S}^2$, the classical Minkowski problem asks for a closed convex surface in $\mathbb{R}^3$ with Gauss curvature $1/ρ(ν)$ at the point with outer normal $ν$. In terms of the support function $h$, it is the Monge--Ampère equation $\det(\nabla^2h+hσ)=ρ$ on $\mathbb{S}^2$, uniquely solvable up to translation when $ρ$ has zero first moment. We solve it by continuation from the sphere: along the linear path $ρ_t=(1-t)+tρ$, the solution $h(t)$ evolves in the artificial time $t$, and its velocity $\partial_th$ solves a linear elliptic equation in divergence form whose kernel consists of the translations. In time we use the explicit Euler or trapezoidal method with step $τ$; in space, spherical harmonics of degree at most $L$ with a Lagrange multiplier for the kernel. For $ρ\in H^{s+2}(\mathbb{S}^2)$ with real $s\ge4$, we prove that all numerical solutions stay strictly convex, with $H^1$ errors $O(τ+L^{-(s+1)})$ for Euler and $O(τ^3+L^{-(s+1)})$ for the trapezoidal scheme, including data quadrature, under independent thresholds for $τ$ and $L$. The proof rests on a priori estimates for the perturbed equation satisfied by each iterate, which keep every step uniformly elliptic, and on the exactly quadratic determinant, which raises the trapezoidal order from two to three. The continuation thus supplies a good initial guess for Newton's method: from the final state it converges quadratically to the Galerkin solution, itself within $O(L^{-(s+1)})$ of the exact solution, whereas started directly from the sphere its first undamped step left the convex cone in our ellipsoid test. Experiments illustrate the rates, use the multiplier as a compatibility diagnostic, reconstruct convex polyhedra from facet normals and areas, and locate umbilics with indices.

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arXiv2026

Sharp Pointwise Error Estimates of Finite Element Methods for the Biharmonic Equation

We prove sharp pointwise error estimates for finite element approximations of the clamped biharmonic equation on convex polygonal domains. For $C^1$-conforming elements, we establish $W^{2,p}$-stability for all $2\le p\le \infty$, together with best-approximation type estimates in $W^{1,\infty}$ and $L^\infty$. For the nonconforming Morley element, we derive optimal estimates in $W^{2,p}$, $W^{1,\infty}$, and $L^\infty$; in particular, the $L^\infty$-error is of order $O(h^2)$ without an additional logarithmic factor. The analysis is based on three ingredients: local energy estimates for conforming and nonconforming plate elements, a removal of the local $L^2$-error term in these estimates, and Hölder estimates for derivatives of the Green's function. These tools are combined with a dyadic decomposition argument for regularized Green's functions. The results remove logarithmic losses present in earlier pointwise estimates for fourth-order problems and yield sharp rates for classical plate elements such as the Argyris, Bogner--Fox--Schmit (BFS), Hsieh--Clough--Tocher (HCT), and Morley elements.

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arXiv2026

Maximal $L^p$-$L^q$ regularity of finite element methods for stochastic parabolic equations in polygonal domains

We prove that a spatial semidiscrete finite element method preserves the stochastic maximal regularity of the heat equation on bounded convex polygonal and polyhedral domains in dimensions two and three. A main difficulty is to establish suitable Green-function estimates for stochastic convolutions, whose paths generally lack classical time derivatives. Besides, the failure of the Burkholder-Davis-Gundy (BDG) inequality at the endpoint $p=\infty$ creates additional difficulties. Our approach leverages a hidden symmetry structure of the discrete Green's function, which enables the use of a key integration-by-parts type formula. Then the discrete stochastic maximal $L^p$-$L^p$ regularity with $p\in [2,\infty)$ is obtained from the discrete local energy estimates and interpolation inequalities. To extend further the discrete stochastic maximal $L^p$-$L^q$ regularity with the full range of integrability exponents, i.e., $p\in (2,\infty), q\in [2,\infty)$ or $p=q=2$, we also show that the discrete convolution kernel satisfies the stochastic Hörmander's condition, allowing the application of a stochastic version of the Calderón-Zygmund theory.

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arXiv2026

UnCut FEM with PHT-splines on unfitted T-meshes

We propose a novel unfitted finite element method based on polynomial splines over hierarchical T-meshes (PHT-splines) for elliptic partial differential equations (PDEs) posed on domains whose boundaries are described implicitly by a level-set function. Exploiting the enhanced smoothness of the spline space, we formulate the method by minimizing the $L^2$ residual of the PDE over an extended neighborhood of the physical domain. This formulation allows the method to be implemented entirely on the background mesh using standard element-wise quadrature. It yields a symmetric positive-definite linear system and avoids the geometric construction of curved cut subregions in boundary elements, as well as the associated nontrivial numerical integration. We establish well-posedness of the numerical scheme and derive optimal \emph{a priori} error estimates in the $H^1$ norm. In addition, we employ a residual-based \emph{a posteriori} error indicator to guide local adaptive refinement of the hierarchical T-meshes. Numerical experiments corroborate the theoretical results and demonstrate the accuracy and efficiency of the proposed approach.

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arXiv2026

A vanishing-viscosity minimizing movement Fourier spectral method for scalar conservation laws

We propose a vanishing-viscosity minimizing movement (VVMM) Fourier spectral method for scalar hyperbolic conservation laws under periodic boundary conditions. The method is a fully discrete space--time residual-minimizing scheme, Fourier spectral in space and polynomial in time, stabilized by vanishing viscosity and heat-kernel postprocessing. A basic analytical difficulty in Fourier discretizations of nonlinear conservation laws is the interaction between Fourier projection and the nonlinear flux; the resulting projection residual contains unresolved high-frequency components. VVMM addresses this issue by selecting, on each time slab, the numerical solution that minimizes {a penalized cut-off viscous conservation laws residual} and then applies heat-kernel postprocessing to damp unresolved modes. We prove a fully discrete $L^1$ error estimate, in any fixed space dimension, that separates the vanishing-viscosity error, the temporal interpolation error, and the spatial Fourier truncation error. {Under exact minimization of the penalized cut-off residual and suitable parameter coupling,} the estimate yields, for each fixed $γ>0$, an $L^1$ convergence rate $N^{-1/2+γ}$, with a constant depending on $γ$. Numerical experiments in one and two dimensions illustrate stable shock capturing without visible Gibbs oscillations and convergence behavior consistent with the theory.

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arXiv2026

High-order mass-, energy- and momentum-conserving methods for the nonlinear Schrödinger equation

This paper introduces a novel formulation and an associated space-time finite element method for simulating solutions to the nonlinear Schrödinger equation. A major advantage of the proposed algorithm is its intrinsic ability to preserve the conservation of mass, energy, and momentum at the discrete level. This is proved for the numerical solutions determined by the fully discrete implicit scheme. An effective iterative scheme is proposed for solving the nonlinear system based on an equivalent formulation which suggests using Newton's iteration for the solution and no iteration for the Lagrange multipliers in the nonlinear system. Extensive numerical examples are provided to demonstrate the high-order convergence and effectiveness of the proposed algorithm in conserving mass, energy, and momentum in the simulation of one-dimensional Ma-solitons and bi-solitons, as well as of two-dimensional solitons governed by the nonlinear Schrödinger equation. The numerical results show that the mass-, energy- and momentum-conserving method designed in this paper also significantly reduces the errors of the numerical solutions in long-time simulations compared with methods which do not conserve these quantities.

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arXiv2026

Dynamic Ritz projection of mean curvature flow and optimal ${\bf L^2}$ convergence of parametric FEM

A new approach is developed to study the convergence of parametric finite element approximations to the mean curvature flow of closed surfaces in three-dimensional space. In this approach, the error analysis is conducted by comparing the numerical solution to a dynamic Ritz projection of the mean curvature flow introduced in this paper, rather than an interpolation of the mean curvature flow, as commonly used in the literature. The errors associated with the dynamic Ritz projection in approximating the mean curvature flow are established in the $L^2$ and $W^{1,p}$ norms. Leveraging these results, optimal-order convergence of parametric finite element methods for mean curvature flow of closed surfaces in the $L^\infty(0,T;L^2)$ norm is proved, including the convergence of parametric finite element methods with piecewise linear finite elements.

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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 $ϕ$-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 $\mathbf{H}^{k,l}_{\boldsymbolβ}$, 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 $\mathbf{H}^{2,2}_{\boldsymbolβ}$.

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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 $(u^0,v^0)\in H^γ(Ω)\times H^{γ-1}(Ω)$, $0<γ\le 1$, 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 $L^2(Ω;Γ)\times H^{-1}(Ω;Γ)$. 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 $h^k+τ^q$ 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

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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