arXiv subjects

Huadong Gao

Publications and source records attributed to Huadong Gao.

6 recordsLinked to original sources

arXiv2026

Optimal Local Error Estimates for Finite Element Methods with Measure-Valued Sources

We study finite element approximations of second-order elliptic problems with measure-valued right-hand sides supported on lower-dimensional sets. The exact solution generally lacks $H^1$-regularity due to the source singularity, which limits global convergence rates of numerical methods. Using a very weak solution framework, we establish well-posedness and global error estimates for standard Lagrange finite element methods on Lipschitz polyhedral/polygonal domains. By using interior estimates techniques, we prove optimal local $L^2$- and $H^1$-error estimates in subdomains that are strictly separated from the support of the measure. Extensive numerical experiments are provided to verify the theoretical results. These results show that for Lagrange FEMs solving elliptic problems with singular right-hand sides, the loss of global convergence is purely local, and that optimal convergence rates still hold away from the singular source.

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arXiv2025

Analysis of A Mixed Finite Element Method for Poisson's Equation with Rough Boundary Data

This paper is concerned with finite element methods for Poisson's equation with rough boundary data. Conventional methods require that the boundary data $g$ of the problem belongs to $H^{1/2} (\partial Ω)$. However, in many applications one has to consider the case when $g$ is in $L^2(\partial Ω)$ only. To this end, very weak solutions are considered to establish the well-posedness of the problem. Most previously proposed numerical methods use regularizations of the boundary data. The main purpose of this paper is to use the Raviart--Thomas mixed finite element method to solve the Poisson equation with rough boundary data directly. We prove that the solution to the proposed mixed method converges to the very weak solution. In particular, we prove that the convergence rate of the numerical solution is $O(h^{1/2})$ in convex domains and $O(h^{s-1/2})$ in nonconvex domains, where $s > 1/2$ depends on the geometry of the domain. The analysis is based on a regularized approach and a rigorous estimate for the corresponding dual problem. Numerical experiments confirm the theoretically predicted convergence rates for the proposed mixed method for Poisson's equation with rough boundary data.

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arXiv2021

The pointwise stabilities of piecewise linear finite element method on non-obtuse tetrahedral meshes of nonconvex polyhedra

Let $Ω$ be a Lipschitz polyhedral (can be nonconvex) domain in $\mathbb{R}^{3}$, and $V_{h}$ denotes the finite element space of continuous piecewise linear polynomials. On non-obtuse quasi-uniform tetrahedral meshes, we prove that the finite element projection $R_{h}u$ of $u \in H^{1}(Ω) \cap C(\overlineΩ)$ (with $R_{h} u$ interpolating $u$ at the boundary nodes) satisfies \begin{align*} \Vert R_{h} u\Vert_{L^{\infty}(Ω)} \leq C \vert \log h \vert \Vert u\Vert_{L^{\infty}(Ω)}. \end{align*} If we further assume $u \in W^{1,\infty}(Ω)$, then \begin{align*} \Vert R_{h} u\Vert_{W^{1, \infty}(Ω)} \leq C \vert \log h \vert \Vert u\Vert_{W^{1, \infty}(Ω)}. \end{align*}

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arXiv2018

A linearized energy preserving finite element method for the dynamical incompressible magnetohydrodynamics equations

We present and analyze a linearized finite element method (FEM) for the dynamical incompressible magnetohydrodynamics (MHD) equations. The finite element approximation is based on mixed conforming elements, where Taylor--Hood type elements are used for the Navier--Stokes equations and Nedelec edge elements are used for the magnetic equation. The divergence free conditions are weakly satisfied at the discrete level. Due to the use of Nedelec edge element, the proposed method is particularly suitable for problems defined on non-smooth and multi-connected domains. For the temporal discretization, we use a linearized scheme which only needs to solve a linear system at each time step. Moreover, the linearized mixed FEM is energy preserving. We establish an optimal error estimate under a very low assumption on the exact solutions and domain geometries. Numerical results which includes a benchmark lid-driven cavity problem are provided to show its effectiveness and verify the theoretical analysis.

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arXiv2017

Error analysis of mixed finite element methods for nonlinear parabolic equations

In this paper, we prove a discrete embedding inequality for the Raviart--Thomas mixed finite element methods for second order elliptic equations, which is analogous to the Sobolev embedding inequality in the continuous setting. Then, by using the proved discrete embedding inequality, we provide an optimal error estimate for linearized mixed finite element methods for nonlinear parabolic equations. Several numerical examples are provided to confirm the theoretical analysis.

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arXiv2016

Analysis of linearized Galerkin-mixed FEMs for the time-dependent Ginzburg--Landau equations of superconductivity

A linearized backward Euler Galerkin-mixed finite element method is investigated for the time-dependent Ginzburg--Landau (TDGL) equations under the Lorentz gauge. By introducing the induced magnetic field $σ = \mathrm{curl} \, {\bf{A}}$ as a new variable, the Galerkin-mixed FE scheme offers many advantages over conventional Lagrange type Galerkin FEMs. An optimal error estimate for the linearized Galerkin-mixed FE scheme is established unconditionally. Analysis is given under more general assumptions for the regularity of the solution of the TDGL equations, which includes the problem in two-dimensional noncovex polygons and certain three dimensional polyhedrons, while the conventional Galerkin FEMs may not converge to a true solution in these cases. Numerical examples in both two and three dimensional spaces are presented to confirm our theoretical analysis. Numerical results show clearly the efficiency of the mixed method, particularly for problems on nonconvex domains.

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