各向异性网格上具混合导数椭圆问题单调离散化的一个反例

任意网格上的单调等高线集无需再距离化

Monotone level-sets on arbitrary meshes without redistancing

本文针对工程实践中使用等高线方法模拟双流体流动时出现的两个问题，提出了解决方案：通过考虑局部网格尺寸改进非均匀网格上的Heaviside函数正则化，以及通过强耦合方法减少再距离化步骤的计算成本和脆弱性。

各向异性椭圆方程的单调$Q^1$有限元方法

A monotone $Q^1$ finite element method for anisotropic elliptic equations

本文针对具有对角占优扩散系数矩阵的各向异性扩散问题，在均匀网格上构造了满足离散最大值原理的单调连续$Q^1$有限元方法，严格证明了其收敛性，并给出了四边形网格上的特定网格约束条件。

An algebraically stabilized method for convection-diffusion-reaction problems with optimal experimental convergence rates on general meshes

Algebraically stabilized finite element discretizations of scalar steady-state convection-diffusion-reaction equations often provide accurate approximate solutions satisfying the discrete maximum principle (DMP). However, it was observed that a deterioration of the accuracy and convergence rates may occur for some problems if meshes without local symmetries are used. The paper investigates these phenomena both numerically and analytically and the findings are used to design a new algebraic stabilization called Symmetrized Monotone Upwind-type Algebraically Stabilized (SMUAS) method. It is prov…

单调拟线性椭圆型偏微分方程的无稳定子弱Galerkin方法

Stabilizer-Free Weak Galerkin Methods for Monotone Quasilinear Elliptic PDEs

本文研究了一类主系数具有梯度非线性的散度型二阶椭圆形边值问题在多边形网格上的无稳定子弱Galerkin方法。通过对非线性系数的某些假设，我们证明了离散问题有唯一解。这是通过证明相关算子满足一定的连续性和单调性来实现的。借助这些性质，我们得到了能量范数下的最优误差估计。我们提出了几个数值例子来验证的误差估计。

具有次临界非线性的单调半线性椭圆型偏微分方程的最优有限元逼近

Optimal finite element approximations of monotone semilinear elliptic PDE with subcritical nonlinearities

本文针对多边形区域上具有次临界增长单调非线性反应项的半线性椭圆边值问题，提出收缩型Picard迭代有限元方法，并利用加权Sobolev空间正则性与Trudinger不等式建立了局部细化网格上的最优先验误差估计。

Two-grid $hp$-version discontinuous Galerkin finite element methods for quasilinear elliptic PDEs on agglomerated coarse meshes

This article considers the extension of two-grid $hp$-version discontinuous Galerkin finite element methods for the numerical approximation of second-order quasilinear elliptic boundary value problems of monotone type to the case when agglomerated polygonal/polyhedral meshes are employed for the coarse mesh approximation. We recall that within the two-grid setting, while it is necessary to solve a nonlinear problem on the coarse approximation space, only a linear problem must be computed on the original fine finite element space. In this article, the coarse space will be constructed by agglome…