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

Analysis of hybridized discontinuous Galerkin methods without elliptic regularity assumptions

In this paper we present new stability and optimal error analyses of hybridized discontinuous Galerkin (HDG) methods which do not require elliptic regularity assumptions. To obtain error estimates without elliptic regularity assumptions, we use new inf-sup conditions based on stabilized saddle point structures of HDG methods. We show that this approach can be applied to obtain optimal error estimates of HDG methods for the Poisson equations, the convection-reaction-diffusion equations, the Stokes equations, and the Oseen equations.

Low-regularity integrators for nonlinear Dirac equations

In this work, we consider the numerical integration of the nonlinear Dirac equation and the Dirac-Poisson system (NDEs) under rough initial data. We propose a ultra low-regularity integrator (ULI) for solving the NDEs which enables optimal first-order time convergence in $H^r$ for solutions in $H^{r}$, i.e., without requiring any additional regularity on the solution. In contrast to classical methods, ULI overcomes the numerical loss of derivatives and is therefore more efficient and accurate for approximating low regular solutions. Convergence theorems and the extension of ULI to second order…

Low Regularity Primal-Dual Weak Galerkin Finite Element Methods for Ill-Posed Elliptic Cauchy Problems

A new primal-dual weak Galerkin (PDWG) finite element method is introduced and analyzed for the ill-posed elliptic Cauchy problems with ultra-low regularity assumptions on the exact solution. The Euler-Lagrange formulation resulting from the PDWG scheme yields a system of equations involving both the primal equation and the adjoint (dual) equation. The optimal order error estimate for the primal variable in a low regularity assumption is established. A series of numerical experiments are illustrated to validate effectiveness of the developed theory.

参数椭圆特征值问题的解析与Gevrey类正则性及其应用

Analytic and Gevrey class regularity for parametric elliptic eigenvalue problems and applications

该研究针对带齐次本质边界条件的参数椭圆特征值问题，通过新型证明技术建立解关于参数的Gevrey类与解析正则性，为相关数值格式收敛性提供理论支撑，可应用于含无穷多参数的随机系数椭圆特征值问题。

将局部Hölder正则化融入PINNs以求解椭圆型PDEs

Incorporating Local Hölder Regularity into PINNs for Solving Elliptic PDEs

本研究将局部Hölder正则化融入PINNs框架，通过构建含该正则项的改进损失函数、提出变距离离散采样策略并建立误差估计，提升了椭圆型PDEs求解的精度与鲁棒性。

A New Primal-Dual Weak Galerkin Method for Elliptic Interface Problems with Low Regularity Assumptions

This article introduces a new primal-dual weak Galerkin (PDWG) finite element method for second order elliptic interface problems with ultra-low regularity assumptions on the exact solution and the interface and boundary data. It is proved that the PDWG method is stable and accurate with optimal order of error estimates in discrete and Sobolev norms. In particular, the error estimates are derived under the low regularity assumption of $u\in H^δ(Ω)$ for $δ> \frac12$ for the exact solution $u$. Extensive numerical experiments are conducted to provide numerical solutions that verify the efficienc…