arXiv subjects

Hengguang Li

Publications and source records attributed to Hengguang Li.

16 recordsLinked to original sources

arXiv2026

Towards $C^0$ finite element methods for fourth-order elliptic equation. Part I: general boundary conditions

This paper is part of a series developing $C^0$ finite element methods for fourth-order elliptic equations on polygonal domains. Here, we investigate how boundary conditions influence the design of effective $C^0$ schemes, specifically focusing on equations without lower-order terms, namely the biharmonic equation. We propose a modified mixed formulation that decomposes the problem into a system of Poisson equations, where the number of equations depends on both the largest interior angle and the boundary conditions on its two adjacent sides. In contrast to the naive mixed formulation, which involves only two Poisson problems, the proposed approach guarantees convergence to the true solution for arbitrary polygonal domains and general boundary conditions, including Navier, Neumann, and mixed boundary conditions. $C^0$ finite element algorithms are developed, rigorous error estimates are established, and numerical experiments are presented to demonstrate the well-posedness and effectiveness of the proposed method.

math.NA↗

arXiv2025

A $C^0$ finite element algorithm for the sixth order problem with simply supported boundary conditions

In this paper, we investigate a sixth order elliptic equation with the simply supported boundary conditions in a polygonal domain. We propose a new method that decouples the sixth order problem into a system of second order equations. Unlike the direct decomposition, which yields three Poisson problems but is restricted to polygonal domains with the largest interior angle no more than $π/{2}$, we rigorously analyze and construct extra Poisson problems to confine the solution into the same function space as that of the original sixth order problem. Consequently, the proposed method can be applied to general polygonal domains. In turn, we also present a $C^0$ finite element algorithm to discretize the new resulting system and establish optimal error estimates for the numerical solution on quasi-uniform meshes. Finally, numerical experiments are performed to validate the theoretical findings.

math.NA↗

arXiv2022

An adaptive finite element method for two-dimensional elliptic equations with line Dirac sources

In this paper, we propose a novel adaptive finite element method for an elliptic equation with line Dirac delta functions as a source term. We first study the well-posedness and global regularity of the solution in the whole domain. Instead of regularizing the singular source term and using the classical residual-based a posteriori error estimator, we propose a novel a posteriori estimator based on an equivalent transmission problem with zero source term and nonzero flux jumps on line fractures. The transmission problem is defined in the same domain as the original problem excluding on line fractures, and the solution is therefore shown to be more regular. The estimator relies on meshes conforming to the line fractures and its edge jump residual essentially uses the flux jumps of the transmission problem on line fractures. The error estimator is proven to be both reliable and efficient, an adaptive finite element algorithm is proposed based on the error estimator and the bisection refinement method. Numerical tests show that quasi-optimal convergence rates are achieved even for high order approximations and the adaptive meshes are only locally refined at singular points.

math.NA↗

arXiv2022

A $C^0$ finite element method for the biharmonic problem with Dirichlet boundary conditions in a polygonal domain

In this paper, we study the biharmonic equation with Dirichlet boundary conditions in a polygonal domain. In particular, we propose a method that effectively decouples the fourth-order problem into a system of two Poison equations and one Stokes equation, or a system of one Stokes equation and one Poisson equation. It is shown that the solution of each system is equivalent to that of the original fourth-order problem on both convex and non-convex polygonal domains. Two finite element algorithms are in turn proposed to solve the decoupled systems. In addition, we show the regularity of the solutions in each decoupled system in both the Sobolev space and the weighted Sobolev space, and we derive the optimal error estimates for the numerical solutions on both quasi-uniform meshes and graded meshes. Numerical test results are presented to justify the theoretical findings.

math.NA↗

arXiv2021

Solving Biharmonic Eigenvalue Problem With Navier Boundary Condition Via Poisson Solvers On Non-Convex Domains

It is well known that the usual mixed method for solving the biharmonic eigenvalue problem by decomposing the operator into two Laplacians may generate spurious eigenvalues on non-convex domains. To overcome this difficulty, we adopt a recently developed mixed method, which decomposes the biharmonic equation into three Poisson equations and still recovers the original solution. Using this idea, we design an efficient biharmonic eigenvalue algorithm, which contains only Poisson solvers. With this approach, eigenfunctions can be confined in the correct space and thereby spurious modes in non-convex domains are avoided. A priori error estimates for both eigenvalues and eigenfunctions on quasi-uniform meshes are obtained; in particular, a convergence rate of $\mathcal{O}({h}^{2α})$ ($ 0<α<π/ω$, $ω> π$ is the angle of the reentrant corner) is proved for the linear finite element. Surprisingly, numerical evidence demonstrates a $\mathcal{O}({h}^{2})$ convergent rate for the quasi-uniform mesh with the regular refinement strategy even on non-convex polygonal domains.

math.NA↗

arXiv2020

A $C^0$ finite element method for the biharmonic problem with Navier boundary conditions in a polygonal domain

In this paper, we study the biharmonic equation with the Navier boundary conditions in a polygonal domain. In particular, we propose a method that effectively decouples the 4th-order problem into a system of Poisson equations. Different from the usual mixed method that leads to two Poisson problems but only applies to convex domains, the proposed decomposition involves a third Poisson equation to confine the solution in the correct function space, and therefore can be used in both convex and non-convex domains. A $C^0$ finite element algorithm is in turn proposed to solve the resulted system. In addition, we derive the optimal error estimates for the numerical solution on both quasi-uniform meshes and graded meshes. Numerical test results are presented to justify the theoretical findings.

math.NA↗

arXiv2020

Regularity and finite element approximation for two-dimensional elliptic equations with line Dirac sources

We study the elliptic equation with a line Dirac delta function as the source term subject to the Dirichlet boundary condition in a two-dimensional domain. Such a line Dirac measure causes different types of solution singularities in the neighborhood of the line fracture. We establish new regularity results for the solution in a class of weighted Sobolev spaces and propose finite element algorithms that approximate the singular solution at the optimal convergence rate. Numerical tests are presented to justify the theoretical findings.

math.NA↗

arXiv2018

A two-grid method for the $C^0$ interior penalty discretization of the Monge-Ampère equation

The purpose of this paper is to analyze an efficient method for the solution of the nonlinear system resulting from the discretization of the elliptic Monge-Ampère equation by a $C^0$ interior penalty method with Lagrange finite elements. We consider the two-grid method for nonlinear equations which consists in solving the discrete nonlinear system on a coarse mesh and using that solution as initial guess for one iteration of Newton's method on a finer mesh. Thus both steps are inexpensive. We give quasi-optimal $W^{1,\infty}$ error estimates for the discretization and estimate the difference between the interior penalty solution and the two-grid numerical solution. Numerical experiments confirm the computational efficiency of the approach compared to Newton's method on the fine mesh.

math.NA↗

arXiv2017

A Novel Stochastic Stratified Average Gradient Method: Convergence Rate and Its Complexity

SGD (Stochastic Gradient Descent) is a popular algorithm for large scale optimization problems due to its low iterative cost. However, SGD can not achieve linear convergence rate as FGD (Full Gradient Descent) because of the inherent gradient variance. To attack the problem, mini-batch SGD was proposed to get a trade-off in terms of convergence rate and iteration cost. In this paper, a general CVI (Convergence-Variance Inequality) equation is presented to state formally the interaction of convergence rate and gradient variance. Then a novel algorithm named SSAG (Stochastic Stratified Average Gradient) is introduced to reduce gradient variance based on two techniques, stratified sampling and averaging over iterations that is a key idea in SAG (Stochastic Average Gradient). Furthermore, SSAG can achieve linear convergence rate of $\mathcal {O}((1-\fracμ{8CL})^k)$ at smaller storage and iterative costs, where $C\geq 2$ is the category number of training data. This convergence rate depends mainly on the variance between classes, but not on the variance within the classes. In the case of $C\ll N$ ($N$ is the training data size), SSAG's convergence rate is much better than SAG's convergence rate of $\mathcal {O}((1-\fracμ{8NL})^k)$. Our experimental results show SSAG outperforms SAG and many other algorithms.

cs.LG↗

arXiv2016

A new anisotropic finite element method on polyhedral domains: interpolation error analysis

Consider the Poisson equation with the Dirichlet boundary condition on a three-dimensional polyhedral domain. For singular solutions from the non-smoothness of the domain boundary, we propose new anisotropic tetrahedral mesh refinement algorithms to improve the convergence of finite element approximation. The proposed algorithm is simple, explicit, and requires less geometric conditions on the mesh and on the domain. Then, we develop interpolation error estimates in suitable weighted spaces for the anisotropic mesh. These estimates can be used to design optimal finite element methods approximating singular solutions. We report numerical test results to validate the method.

math.NA↗

arXiv2016

Regularity and a priori error analysis of a Ventcel problem in polyhedral domains

We consider the regularity of a mixed boundary value problem for the Laplace operator on a polyhedral domain, where Ventcel boundary conditions are imposed on one face of the polyhedron and Dirichlet boundary conditions are imposed on the complement of that face in the boundary. We establish improved regularity estimates for the trace of the variational solution on the Ventcel face, and use them to derive a decomposition of the solution into a regular and a singular part that belongs to suitable weighted Sobolev spaces. This decomposition, in turn, via interpolation estimates both in the interior as well as on the Ventcel face, allows us to perform an a priori error analysis for the Finite Element approximation of the solution on anisotropic graded meshes. Numerical tests support the theoretical analysis.

math.AP↗

arXiv2016

Differential operators on domains with conical points: precise uniform regularity estimates

We study families of strongly elliptic, second order differential operators with singular coefficients on domains with conical points. We obtain uniform estimates on their inverses and on the regularity of the solutions to the associated Poisson problem with mixed boundary conditions. The coefficients and the solutions belong to (suitable) weighted Sobolev spaces. The space of coefficients is a Banach space that contains, in particular, the space of smooth functions. Hence, our results extend classical well-posedness results for strongly elliptic equations in domains with conical points to problems with singular coefficients. We furthermore provide precise uniform estimates on the norms of the solution operators.

math.AP↗

arXiv2012

Uniform shift estimates for transmission problems and optimal rates of convergence for the parametric Finite Element Method

Let $Ω\subset \RR^d$, $d \geqslant 1$, be a bounded domain with piecewise smooth boundary $\partial Ω$ and let $U$ be an open subset of a Banach space $Y$. Motivated by questions in "Uncertainty Quantification," we consider a parametric family $P = (P_y)_{y \in U}$ of uniformly strongly elliptic, second order partial differential operators $P_y$ on $Ω$. We allow jump discontinuities in the coefficients. We establish a regularity result for the solution $u: Ω\times U \to \RR$ of the parametric, elliptic boundary value/transmission problem $P_y u_y = f_y$, $y \in U$, with mixed Dirichlet-Neumann boundary conditions in the case when the boundary and the interface are smooth and in the general case for $d=2$. Our regularity and well-posedness results are formulated in a scale of broken weighted Sobolev spaces $\hat\maK^{m+1}_{a+1}(Ω)$ of Babuška-Kondrat'ev type in $Ω$, possibly augmented by some locally constant functions. This implies that the parametric, elliptic PDEs $(P_y)_{y \in U}$ admit a shift theorem that is uniform in the parameter $y\in U$. In turn, this then leads to $h^m$-quasi-optimal rates of convergence (i.e. algebraic orders of convergence) for the Galerkin approximations of the solution $u$, where the approximation spaces are defined using the "polynomial chaos expansion" of $u$ with respect to a suitable family of tensorized Lagrange polynomials, following the method developed by Cohen, Devore, and Schwab (2010).

math.NA↗

arXiv2012

Axisymmetric Stokes equations in polygonal domains: regularity and finite element approximations

We study the regularity and finite element approximation of the axisymmetric Stokes problem on a polygonal domain $Ω$. In particular, taking into account the singular coefficients in the equation and non-smoothness of the domain, we establish the well-posedness and full regularity of the solution in new weighted Sobolev spaces $\maK^m_{μ, 1}(Ω)$. Using our a priori results, we give a specific construction of graded meshes on which the Taylor-Hood mixed method approximates singular solutions at the optimal convergence rate. Numerical tests are presented to confirm the theoretical results in the paper.

math.NA↗

arXiv2012

Analysis of Schrödinger operators with inverse square potentials {II}: FEM and approximation of eigenfunctions in the periodic case

Let $V$ be a {\em periodic} potential on $\RR^3$ that is smooth everywhere except at a discrete set $\maS$ of points, where it has singularities of the form $Z/ρ^2$, with $ρ(x) = |x - p|$ for $x$ close to $p$ and $Z$ is continuous, $Z(p) > -1/4$ for $p \in \maS$. We also assume that $ρ$ and $Z$ are smooth outside $\maS$ and $Z$ is smooth in polar coordinates around each singular point. Let us denote by $Λ$ the periodicity lattice and set $\TT := \RR^3/ Λ$. In the first paper of this series \cite{HLNU1}, we obtained regularity results in weighted Sobolev space for the eigenfunctions of the Schrödinger-type operator $H = -Δ+ V$ acting on $L^2(\TT)$, as well as for the induced $\vt k$--Hamiltonians $\Hk$ obtained by resticting the action of $H$ to Bloch waves. In this paper we present two related applications: one to the Finite Element approximation of the solution of $(L+\Hk) v = f$ and one to the numerical approximation of the eigenvalues, $λ$, and eigenfunctions, $u$, of $\Hk$. We give optimal, higher order convergence results for approximation spaces defined piecewise polynomials. Our numerical tests are in good agreement with the theoretical results.

math-ph↗

arXiv2012

Analysis of Schrödinger operators with inverse square potentials I: regularity results in 3D

Let $V$ be a potential on $\RR^3$ that is smooth everywhere except at a discrete set $\maS$ of points, where it has singularities of the form $Z/ρ^2$, with $ρ(x) = |x - p|$ for $x$ close to $p$ and $Z$ continuous on $\RR^3$ with $Z(p) > -1/4$ for $p \in \maS$. Also assume that $ρ$ and $Z$ are smooth outside $\maS$ and $Z$ is smooth in polar coordinates around each singular point. We either assume that $V$ is periodic or that the set $\maS$ is finite and $V$ extends to a smooth function on the radial compactification of $\RR^3$ that is bounded outside a compact set containing $\maS$. In the periodic case, we let $Λ$ be the periodicity lattice and define $\TT := \RR^3/ Λ$. We obtain regularity results in weighted Sobolev space for the eigenfunctions of the Schrödinger-type operator $H = -Δ+ V$ acting on $L^2(\TT)$, as well as for the induced $\vt k$--Hamiltonians $\Hk$ obtained by restricting the action of $H$ to Bloch waves. Under some additional assumptions, we extend these regularity and solvability results to the non-periodic case. We sketch some applications to approximation of eigenfunctions and eigenvalues that will be studied in more detail in a second paper.

math.NA↗