Local Finite Element Approximation of Sobolev Differential Forms

有限差分方案用于Sobolev空间中的随机偏微分方程

Finite difference schemes for stochastic partial differential equations in Sobolev spaces

本文研究了近似抛物型随机偏微分方程的有限差分方案的L_p估计，探讨了初始条件和自由项在Sobolev空间中的误差展开及Richardson方法加速近似。

Lagrangian finite elements in Sobolev-like spaces of order $3/2$

This paper introduces a Sobolev-like space of order $3/2$, denoted as $\widehat{H}^{3/2}$, for Lagrangian finite elements, especially for $C^0$ elements. It is motivated by the limitations of current stability analysis of the evolving surface finite element method (ESFEM), which relies exclusively on an energy estimate framework. To establish a PDE-based analysis framework for ESFEM, we encounter a fundamental regularity mismatch: the ESFEM adopts the $C^0$ elements, while the PDE regularity theory requires $H^{3/2}$ regularity for solutions. To overcome this difficulty, we first examine the p…

Efficient Direct Space-Time Finite Element Solvers for Parabolic Initial-Boundary Value Problems in Anisotropic Sobolev Spaces

We consider a space-time variational formulation of parabolic initial-boundary value problems in anisotropic Sobolev spaces in combination with a Hilbert-type transformation. This variational setting is the starting point for the space-time Galerkin finite element discretization that leads to a large global linear system of algebraic equations. We propose and investigate new efficient direct solvers for this system. In particular, we use a tensor-product approach with piecewise polynomial, globally continuous ansatz and test functions. The developed solvers are based on the Bartels-Stewart met…

通过带有微分算子和边界算子的格林核方法构造索伯列夫空间的再生核

Reproducing Kernels of Sobolev Spaces via a Green Kernel Approach with Differential Operators and Boundary Operators

本文引入向量微分算子P和边界算子B，利用格林核方法构造与经典索伯列夫空间嵌入的再生核及其希尔伯特空间，并给出了内积表示、特征函数与特征值关系以及级数展开。

基于核方法的随机偏微分方程求解

Kernel-based Methods for Stochastic Partial Differential Equations

本文通过结合无网格逼近和克里金插值技术，将核方法从确定性数据扩展到随机数据，用于求解高维随机偏微分方程，并通过数值实验验证了不同核函数下近似概率分布的适定性。

Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces

We introduce a novel spectral, finite-dimensional approximation of general Sobolev spaces in terms of Chebyshev polynomials. Based on this polynomial surrogate model (PSM), we realise a variational formulation, solving a vast class of linear and non-linear partial differential equations (PDEs). The PSMs are as flexible as the physics-informed neural nets (PINNs) and provide an alternative for addressing inverse PDE problems, such as PDE-parameter inference. In contrast to PINNs, the PSMs result in a convex optimisation problem for a vast class of PDEs, including all linear ones, in which case …

负Sobolev空间上的插值算子

Interpolation Operator on negative Sobolev Spaces

本文提出一种Scott-Zhang型投影算子，用于任意多项式阶的拉格朗日元。该算子在负范数下稳定，并具有最优收敛速率，用于抛物问题的插值误差估计和最小二乘有限元法中平滑粗糙右端项。

有限元近似在Sobolev不等式极值函数中的收敛率

Convergence rate of the finite element approximation for extremizers of Sobolev inequalities

本文研究了有限元数值方案逼近Sobolev不等式极值函数的收敛率，证明了在二维多边形凸域中，有限元解在$ L^2 $和$ H^1 $范数下分别以$ O(h^2) $和$ O(h) $的速率收敛，其中$ h $是域三角剖分的网格尺寸。

加权Sobolev空间上微分方程的数值分析：超越经典正交多项式

Numerical Analysis of differential equations on weighted Sobolev spaces: beyond classical orthogonal polynomials

本文为加权Sobolev空间上的微分方程数值分析奠定数学基础，通过分解主导线性分量实现基于正交多项式的求解，并提出新的Sobolev正交多项式概念，应用于Gross-Pitaevskii方程和随机共振的严格证明。

Weighted Sobolev regularity and rate of approximation of the obstacle problem for the integral fractional Laplacian

We obtain regularity results in weighted Sobolev spaces for the solution of the obstacle problem for the integral fractional Laplacian. The weight is a power of the distance to the boundary. These bounds then serve us as a guide in the design and analysis of an optimal finite element scheme over graded meshes.

分数Sobolev常数的Galerkin逼近

Galerkin Approximation of the Fractional Sobolev Constant

研究分数Sobolev不等式在不同维度下的离散最优常数估计，采用线性元Galerkin逼近方法，在单位球内建立收敛率。