Claire Chainais-Hillairet, Maxime Herda
In this work, we apply an iterative energy method à la de Giorgi in order to establish $L^{\infty}$ bounds for numerical solutions of noncoercive convection-diffusion equations with mixed Dirichlet-Neumann boundary conditions.
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Maximum principle in linear finite element approximations of anisotropic diffusion-convection-reaction problems
针对各向异性扩散-对流-反应问题的线性有限元近似，提出一种网格条件以保证离散最大值原理，该条件要求网格为单纯形且在扩散矩阵逆指定的度量下满足特定非钝角条件，二维情形可弱化为广义Delaunay条件的摄动。
We devised a first order time stepping ensemble hybridizable discontinuous Galerkin (HDG) method for a group of parameterized convection diffusion PDEs with different initial conditions, body forces, boundary conditions and coefficients in our earlier work [3]. We obtained an optimal convergence rate for the ensemble solutions in $L^\infty(0,T;L^2(Ω))$ on a simplex mesh; and obtained a superconvergent rate for the ensemble solutions in $L^2(0,T;L^2(Ω))$ after an element-by-element postprocessing if polynomials degree $k\ge 1$ and the coefficients of the PDEs are independent of time. In this wo…
A mass-preserving two-step Lagrange-Galerkin scheme of second order in time for convection-diffusion problems is presented, and convergence with optimal error estimates is proved in the framework of $L^2$-theory. The introduced scheme maintains the advantages of the Lagrange-Galerkin method, i.e., CFL-free robustness for convection-dominated problems and a symmetric and positive coefficient matrix resulting from the discretization. In addition, the scheme conserves the mass on the discrete level if the involved integrals are computed exactly. Unconditional stability and error estimates of seco…
An Analysis of the Weak Finite Element Method for Convection-Diffusion Equations
本文研究了弱有限元方法求解对流-扩散方程，推导了最优阶误差估计，并给出了H^1超收敛性。
This paper develops a two-level fourth-order scheme for solving time-fractional convection-diffusion-reaction equation with variable coefficients subjected to suitable initial and boundary conditions. The basis properties of the new approach are investigated and both stability and error estimates of the proposed numerical scheme are deeply analyzed in the $L^{\infty}(0,T;L^{2})$-norm. The theory indicates that the method is unconditionally stable with convergence of order $O(k^{2-\fracλ{2}}+h^{4})$, where $k$ and $h$ are time step and mesh size, respectively, and $λ\in(0,1)$. This result sugge…
Numerical analysis of a stabilized scheme for an optimal control problem governed by a parabolic convection--diffusion equation
本文研究了由抛物型对流-扩散方程支配的最优控制问题，提出稳定化方案并推导了误差估计，通过数值实验验证了其收敛性。
In this article we consider the numerical approximation of the convection-diffusion-reaction equation. One of the main challenges of designing a numerical method for this problem is that boundary layers occurring in the convection-dominated case can lead to non-physical oscillations in the numerical approximation, often referred to as Gibbs phenomena. The idea of this article is to consider the approximation problem as a residual minimization in dual norms in Lq-type Sobolev spaces, with 1 < q < $\infty$. We then apply a non-standard, non-linear PetrovGalerkin discretization, that is applicabl…
Error Estimates for First- and Second-Order Lagrange-Galerkin Moving Mesh Schemes for the One-Dimensional Convection-Diffusion Equation
本文针对一维对流扩散方程提出基于 Lagrange-Galerkin 方法的移动网格格式，证明一阶和二阶时间离散下的最优误差估计及质量守恒，并通过数值实验验证其能克服固定网格方法的局限。
Radial basis function methods for optimal control of the convection-diffusion equation
本文提出两种局部径向基函数方法，用于解决对流-扩散控制问题，通过预处理技术缓解病态问题，提高计算效率。
Efficient and simple fourth-order compact finite difference methods for convection-diffusion-reaction equations on arbitrary curved domains
本文提出了一种高效的四阶紧致有限差分方法，用于求解具有变量光滑系数的二维对流-扩散-反应方程及其在复杂、细长和曲域上的Dirichlet边界条件，该方法在任意不规则 stencil 中保持了简单且高阶的精度。
Fourth-Order Compact FDMs for Steady and Time-Dependent Nonlinear Convection-Diffusion Equations
针对稳态与非稳态非线性对流-扩散方程，提出低污染的二阶至四阶紧致九点 FDM，并证明 M 矩阵性质与离散最大值原理，数值精度显著优于 DG。
An Efficient Second-Order-in-Time Penalty-Projection Ensemble Eddy Viscosity Method for Parameterized Navier-Stokes Flows
针对参数化纳维-斯托克斯流，提出时间二阶的惩罚-投影集合涡粘性算法，该算法基于BDF-2，具备grad-div稳定，经数值实验验证其收敛性与有效性。
In this paper, based on the developed nonlinear fourth-order operator and method of order reduction, a novel fourth-order compact difference scheme is constructed for the mixed-type time-fractional Burgers' equation, from which $L_1$-discretization formula is employed to deal with the terms of fractional derivative, and the nonlinear convection term is discretized by nonlinear compact difference operator. Then a fully discrete compact difference scheme can be established by approximating spatial second-order derivative with classic compact difference formula. The convergence and stability are …
A Second Order Energy Stable Scheme for the Cahn-Hilliard-Hele-Shaw Equations
本文提出一种二阶时间有限差分方案，用于Cahn-Hilliard-Hele-Shaw方程，该方法唯一可解且无条件能量稳定，通过非线性多网格求解器高效求解非线性方程组，并通过误差估计分析获得最优收敛率。
Time-space adaptive discontinuous Galerkin method for advection-diffusion equations with non-linear reaction mechanism
本文提出一种基于椭圆重构技术的时-空自适应间断广义有限元方法，用于求解非线性反应机制主导的对流主导抛物型问题，通过后验误差估计器提升计算效率。
A robust time-split linearized explicit/implicit technique for two-dimensional hydrodynamic model: an application to floods in Cameroon far north region
本文提出一种时间分裂显式/隐式技术求解二维水动力模型，具有四阶空间精度和二阶时间收敛性，并成功应用于喀麦隆极北地区洪水模拟。
A Reynolds-semi-robust and pressure robust Hybrid High-Order method for the time dependent incompressible Navier--Stokes equations on general meshes
本研究针对不可压缩Navier--Stokes方程，提出一种Reynolds半鲁棒、压力鲁棒的混合高阶（HHO）离散格式，通过对流项子网格投影惩罚项实现半鲁棒性，收敛阶达$h^{k+\frac12}$，经二维多边形网格数值验证。
Unified convergence analysis of numerical schemes for a miscible displacement problem
本文统一分析了多种数值方法在不可压缩流体置换模型中的收敛性，通过梯度离散化方法框架，提出在最小正则性假设下，浓度在$ L^\infty(0,T; L^2(Ω))$中的收敛性结果，并比较了两种方案在不同条件下的性能。
In this paper, we propose and analyze a temporally second-order accurate, fully discrete finite element method for the magnetohydrodynamic (MHD) equations. A modified Crank--Nicolson method is used to discretize the model and appropriate semi-implicit treatments are applied to the fluid convection term and two coupling terms. These semi-implicit approximations result in a linear system with variable coefficients for which the unique solvability can be proved theoretically. In addition, we use a decoupling projection method of the Van Kan type \cite{vankan1986} in the Stokes solver, which compu…
A cross-diffusion system modelling rivaling gangs: global existence of bounded solutions and FCT stabilization for numerical simulation
本文研究敌对帮派交叉扩散模型，证明小初值下全局有界经典解及其向齐次稳态收敛，并用保正的 FEM-FCT 数值揭示扩散或对流主导下的不同渐近行为。
Asymptotic-numerical study of supersensitivity for generalized Burgers equations
本文针对广义Burgers方程，研究边界扰动对内部过渡层位置的超灵敏影响，提出计算稳态过渡层位置的算法，该算法可推广至凸非线性粘性守恒律并支持并行计算。
Numerical schemes for the solution of the Euler equations have recently been developed, which involve the discretisation of the internal energy equation, with corrective terms to ensure the correct capture of shocks, and, more generally, the consistency in the Lax-Wendroff sense. These schemes may be staggered or colocated, using either struc-tured meshes or general simplicial or tetrahedral/hexahedral meshes. The time discretization is performed by fractional-step algorithms; these may be either based on semi-implicit pressure correction techniques or segregated in such a way that only explic…
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