Low Regularity Estimates for CutFEM Approximations of an Elliptic Problem with Mixed Boundary Conditions

CutFEM Based on Extended Finite Element Spaces

We develop a general framework for construction and analysis of discrete extension operators with application to unfitted finite element approximation of partial differential equations. In unfitted methods so called cut elements intersected by the boundary occur and these elements must in general by stabilized in some way. Discrete extension operators provides such a stabilization by modification of the finite element space close to the boundary. More precisely, the finite element space is extended from the stable interior elements over the boundary in a stable way which also guarantees optima…

Analysis of optimal preconditioners for CutFEM

In this paper we consider a class of unfitted finite element methods for scalar elliptic problems. These so-called CutFEM methods use standard finite element spaces on a fixed unfitted triangulation combined with the Nitsche technique and a ghost penalty stabilization. As a model problem we consider the application of such a method to the Poisson interface problem. We introduce and analyze a new class of preconditioners that is based on a subspace decomposition approach. The unfitted finite element space is split into two subspaces, where one subspace is the standard finite element space assoc…

一种用于椭圆界面问题非适应有限元离散化的多网格方法

A Multigrid Method for Unfitted Finite Element Discretizations of Elliptic Interface Problems

本文提出一种基于新型细化算子和界面平滑器的多网格方法，用于解决线性非适应有限元中的离散泊松界面问题，展示其在大扩散系数跃变时的鲁棒性。

带边界修正的切割有限元方法的后验误差估计

A Posteriori Error Estimates with Boundary Correction for a Cut Finite Element Method

本文为CutFEM方法提出带边界修正的后验误差估计，考虑非多边形边界及几何与数据逼近，理论证明可靠性与效率，且常数对网格切割方式鲁棒。

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.

用于混合维域的对流扩散问题的切割有限元方法

Cut Finite Element Methods for Convection-Diffusion in Mixed-Dimensional Domains

本文提出一种切割有限元方法，用于混合维域中的对流扩散问题，通过混合维方向导数和散度算子抽象表达问题，采用固定背景网格和主动网格表示每个流形组件，证明了能量和L2范数的先验误差估计，并通过数值实验验证了收敛性。

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。

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.

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…

Regularity for quasilinear vectorial elliptic systems through an iterative scheme with numerical applications

We consider an iterative procedure to solve quasilinear elliptic systems with $p$-growth. The scheme was first considered by Koshelev in the quadratic case $p=2$. We present numerical applications as well as applications to higher regularity properties.

二阶椭圆偏微分方程在谱Barron空间中的正则性

Regularity of Second-Order Elliptic PDEs in Spectral Barron Spaces

本文研究了二阶椭圆偏微分方程在谱Barron空间中的正则性，证明了在弱椭圆性和小性假设下解的正则性提升，并推导出可由独立于空间维度的神经网络近似的一类PDEs。