Hans-Goerg Roos, Martin Schopf
Consider a singularly perturbed convection-diffusion problem with small, variable diffusion. Based on certain a priori estimates for the solution we prove robustness of a finite element method on a Duran-Shishkin mesh.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Variational Multiscale Proper Orthogonal Decomposition: Convection-Dominated Convection-Diffusion Equations
针对对流主导方程的本征正交分解降阶模型，提出一种变分多尺度闭合建模策略，通过有限元离散分析提高数值精度并展示理论收敛率。
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.
Unconditionally stable schemes for non-stationary convection-diffusion equations
针对非定常对流扩散方程，通过引入新变量和反应算子分裂的显隐式逼近，构造了无条件稳定的数值格式。
Some remarks on strongly coupled systems of convection-diffusion equations in 2D
研究二维强耦合对流-扩散方程组解的层结构，提出特殊情况下的一些初步结果。
We consider the numerical approximation of the ill-posed data assimilation problem for stationary convection-diffusion equations and extend our previous analysis in [Numer. Math. 144, 451--477, 2020] to the convection-dominated regime. Slightly adjusting the stabilized finite element method proposed for dominant diffusion, we draw upon a local error analysis to obtain quasi-optimal convergence along the characteristics of the convective field through the data set. The weight function multiplying the discrete solution is taken to be Lipschitz and a corresponding super approximation result (disc…
We present a robust a posteriori error estimator for the weak Galerkin finite element method applied to stationary convection-diffusion equations in the convection-dominated regime. The estimator provides global upper and lower bounds of the error %measured in a suitable norm and is robust in the sense that upper and lower bounds are uniformly bounded with respect to the diffusion coefficient. Results of the numerical experiments are presented to illustrate the performance of the error estimator.
A finite element method of any order is applied on a Bakhvalov-type mesh to solve a singularly perturbed convection--diffusion equation in 2D, whose solution exhibits exponential boundary layers. A uniform convergence of (almost) optimal order is proved by means of a carefully defined interpolant.
A robust solver for H(curl) convection-diffusion and its local Fourier analysis
针对二维H(curl)对流扩散问题，提出基于指数拟合离散的鲁棒高效多重网格求解器，设计含顺风型Gauss-Seidel与辅助问题光滑的混合光滑器，用LFA分析收敛性，验证了算法在两类主导情形下的鲁棒性。
Edge-averaged virtual element methods for convection-diffusion and convection-dominated problems
本文开发了边平均虚拟元（EAVE）方法解决对流占优的对流扩散问题，引入了确保单调性的变体及与泊松方程最低阶虚拟元刚度矩阵集成的综合框架，数值实验验证了其最优收敛率。
A Posteriori Error Estimates for Non-Stationary Non-Linear Convection-Diffusion Equations
本文基于随机对流扩散问题，推导非稳非线性对流扩散方程的后验误差估计，探讨扩散、对流、反应和非线性（噪声）参数的相互作用。
Considering a singularly perturbed convection-diffusion problem, we present an analysis for a superconvergence result using pointwise interpolation of Gauß-Lobatto type for higher-order streamline diffusion FEM. We show a useful connection between two different types of interpolation, namely a vertex-edge-cell interpolant and a pointwise interpolant. Moreover, different postprocessing operators are analysed and applied to model problems.
We report in this paper the analysis for the linear and nonlinear version of the flux corrected transport (FEM-FCT) scheme in combination with the backward Euler time-stepping scheme applied to time-dependent convection-diffusion-reaction problems. We present the stability and error estimates for the linear and nonlinear FEM-FCT scheme. Numerical results confirm the theoretical predictions.
This article devises a new primal-dual weak Galerkin finite element method for the convection-diffusion equation. Optimal order error estimates are established for the primal-dual weak Galerkin approximations in various discrete norms and the standard $L^2$ norms. A series of numerical experiments are conducted and reported to verify the theoretical findings.
This paper is devoted to the construction and analysis of the finite element approximations for the $H(D)$ convection-diffusion problems, where $D$ can be chosen as ${\rm grad}$, ${\rm curl}$ or ${\rm div}$ in 3D case. An essential feature of these constructions is to properly average the PDE coefficients on the sub-simplexes. The schemes are of the class of exponential fitting methods that result in special upwind schemes when the diffusion coefficient approaches to zero. Their well-posedness are established for sufficiently small mesh size assuming that the convection-diffusion problems are …
A convection-diffusion problem with a large shift on Duran meshes
研究空间大位移对流-扩散问题，在适配层及弱层处较粗的Duran型网格上分析高阶有限元方法，并通过数值实验验证理论结果。
An abstract framework is developed that enables the analysis of algebraically stabilized discretizations in a unified way. This framework is applied to a discretization of this kind for convection-diffusion-reaction equations. The definition of this scheme contains a new limiter that improves a standard one in such a way that local and global discrete maximum principles are satisfied on arbitrary simplicial meshes.
The Heterogeneous Multiscale Finite Volume Method for Convection-Diffusion-Reaction Problem
本文提出基于异质多尺度方法的有限体积法，用于多尺度对流-扩散-反应问题，证明在周期性介质中H^1范数下的最优收敛阶。
A modified dynamic diffusion finite element method with optimal convergence rate for convection-diffusion-reaction equations
本文提出一种修正的非线性动态扩散有限元方法求解对流-扩散-反应方程，无需稳定化参数且能抑制虚假振荡，并证明了离散解的存在唯一性及能量范数下最优一阶收敛速率。
A second-order numerical method for two-dimensional two-sided space fractional convection diffusion equation
本文提出一种交替方向隐式方法，用于求解二维双侧空间分数对流扩散方程，证明了该差分格式在空间和时间方向上均具无条件von Neumann稳定性及二阶收敛性。
A Flexible Velocity Boltzmann Scheme for Convection-Diffusion Equations
本文提出一种速度可调的有限速度Boltzmann框架，结合通量差分裂与广义动力学Lax-Wendroff格式及限制器，有效控制数值扩散并求解一维、二维非线性对流-扩散方程。
The objective of this contribution is to develop a convergence analysis for SUPG-stabilized Virtual Element Methods in diffusion-convection problems that is robust also in the convection dominated regime. For the original method introduced in [Benedetto et al, CMAME 2016] we are able to show an "almost uniform" error bound (in the sense that the unique term that depends in an unfavorable way on the parameters is damped by a higher order mesh-size multiplicative factor). We also introduce a novel discretization of the convection term that allows us to develop error estimates that are fully robu…
Matrix-equation-based strategies for convection-diffusion equations
本文提出基于矩阵方程的预处理方法，用于求解分离系数和主导对流的对流扩散方程离散后的非对称线性系统，通过矩阵方程显式解替代线性求解，展示方法的有效性。
We consider the a posteriori error estimation for convection-diffusion-reaction equations in both diffusion-dominated and convection/reaction-dominated regimes. We present an explicit hybrid estimator, which, in each regime, is proved to be reliable and efficient with constants independent of the parameters in the underlying problem. For convection-dominated problems, the norm introduced by Verf{ü}rth \cite{verf2005confusion} is used to measure the approximation error. Various numerical experiments are performed to (1) demonstrate the robustness of the hybrid estimator; (2) show that the hybri…
A Unified Analysis of Nonconforming Virtual Element Methods for Convection Diffusion Reaction Problem
本文研究了非顺应虚拟元方法在对流占优问题中的稳定性与收敛性，提出新方法并讨论了计算可行性。
The adaptive Crouzeix-Raviart element method for convection-diffusion eigenvalue problems
本文研究了对流扩散本征值问题的自适应Crouzeix-Raviart非协调元方法，提出后验误差估计器并验证其可靠性和有效性，数值结果证实了算法的有效性。
A high-order finite element method is proposed to solve the nonlinear convection-diffusion equation on a time-varying domain whose boundary is implicitly driven by the solution of the equation. The method is semi-implicit in the sense that the boundary is traced explicitly with a high-order surface-tracking algorithm, while the convection-diffusion equation is solved implicitly with high-order backward differentiation formulas and fictitious-domain finite element methods. By two numerical experiments for severely deforming domains, we show that optimal convergence orders are obtained in energy…
Numerical methods for convection-diffusion problems or The 30 years war
本文综述了对流-扩散问题的数值方法，探讨了其历史背景及最新发展，强调了数值技术的核心原理和未解难题。
We present recent finite element numerical results on a model convection-diffusion problem in the singular perturbed case when the convection term dominates the problem. We compare the standard Galerkin discretization using the linear element with a saddle point least square discretization that uses quadratic test functions, trying to control and explain the non-physical oscillations of the discrete solutions. We also relate the up-winding Petrov-Galerkin method and the stream-line diffusion discretization method, by emphasizing the resulting linear systems and by comparing appropriate error n…
In this article, using the weighted discrete least-squares, we propose a patch reconstruction finite element space with only one degree of freedom per element. As the approximation space, it is applied to the discontinuous Galerkin methods with the upwind scheme for the steady-state convection-diffusion-reaction problems over polytopic meshes. The optimal error estimates are provided in both diffusion-dominated and convection-dominated regimes. Furthermore, several numerical experiments are presented to verify the theoretical error estimates, and to well approximate boundary layers and/or inte…
Adaptive discontinuous Galerkin methods for non-linear diffusion-convection-reaction equations
本文提出自适应不连续伽辽金方法用于求解对流占优的非线性扩散-对流-反应方程，通过矩阵重排方案设计高效预处理器，有效抑制伪振荡并准确捕捉尖锐层。
A note of "pointwise estimates of the SDFEM for convection--diffusion problems with characteristic layers"
针对特征层问题，在Shishkin网格上对流线扩散有限元方法（SDFEM）提出了有用的逐点误差估计。
A new local discontinuous Galerkin (LDG) method for convection-diffusion equations on overlapping meshes with periodic boundary conditions was introduced in \cite{Overlap1}. With the new method, the primary variable $u$ and the auxiliary variable $p=u_x$ are solved on different meshes. In this paper, we will extend the idea to convection-diffusion equations with non-periodic boundary conditions, i.e. Neumann and Dirichlet boundary conditions. The main difference is to adjust the boundary cells. Moreover, we study the stability and suboptimal error estimates. Finally, numerical experiments are …
CIP-stabilized Virtual Elements for diffusion-convection-reaction problems
本文针对扩散-对流-反应问题，提出采用连续内部惩罚（CIP）稳定化的虚拟元方法，以在对流占优情形下获得准鲁棒格式，并通过数值实验验证了理论分析结果。
In this paper we analyze a lowest order virtual element method for the load classic reaction-convection-diffusion problem and the convection-diffusion spectral problem, where the assumptions on the polygonal meshes allow to consider small edges for the polygons. Under well defined seminorms depending on a suitable stabilization for this geometrical approach, we derive the well posedness of the numerical scheme and error estimates for the load problem, whereas for the spectral problem we derive convergence and error estimates for the eigenvalues and eigenfunctions. We report numerical tests to …
In this work, we propose a new quasi-optimal test norm for a discontinuous Petrov-Galerkin (DPG) discretization of the ultra-weak formulation of the convection-diffusion equation. We prove theoretically that the proposed test norm leads to bounds between the target norm and the energy norm induced by the test norm which are robust with respect to the diffusion parameter in the solution and gradient components and have favorable scalings in the trace components. We conclude with numerical experiments to confirm our theoretical results.
We consider a singularly perturbed convection-diffusion problem that has in addition a shift term. We show a solution decomposition using asymptotic expansions and a stability result. Based upon this we provide a numerical analysis of high order finite element method on layer adapted meshes. We also apply a new idea of using a coarser mesh in places where weak layers appear. Numerical experiments confirm our theoretical results.
Moving Mesh with Streamline Upwind Petrov-Galerkin (MM-SUPG) Method for Time-dependent Convection-Dominated Convection-Diffusion Problems
本文提出一种结合残差基 SUPG 稳定化与基于度量的移动网格 PDE 方法的 MM-SUPG 方法，通过网格适应与稳定化相互作用提升对流占优问题的求解精度与稳定性。
The discontinuous Galerkin method for fractional degenerate convection-diffusion equations
针对分数阶扩散（Lévy）算子扰动的强退化对流扩散方程，提出并研究不连续伽辽金方法，证明稳定性估计和收敛性结果，并通过数值实验展示解的定性行为。
A Matlab Tutorial for Diffusion-Convection-Reaction Equations using DGFEM
本文介绍利用不连续 Galerkin 有限元方法求解稳态扩散-对流-反应方程的 MATLAB 程序，采用稀疏矩阵和向量化技术提高效率。
Explaining oscillatory behavior in convection-diffusion discretization
本文研究了对流-扩散离散化中的振荡现象，提出消除振荡的方法，并介绍了新的误差分析方法，用于构建多维对流占优问题的稳健离散化方案。
We present a Scharfetter-Gummel (SG) stabilization scheme for high-order Hybrid Discontinuous Galerkin (HDG) approximations of convection-diffusion problems. The scheme is based on a careful choice of the stabilization parameters used to define the numerical flux in the HDG method. We show that, in one dimension, the SG-HDG scheme is equivalent to the Finite Volume method stabilized with the Scharfetter--Gummel on the dual grid, for all orders of HDG schemes.
A hydrogeological model for the spread of pollution in an aquifer is considered. The model consists in a convection-diffusion-reaction equation involving the dispersion tensor which depends nonlinearly of the fluid velocity. We introduce an explicit flux in the model and use a mixed Finite Element Method for the discretization. We provide existence, uniqueness and stability results for the discrete model. A convergence result is obtained for the semi-discretized in time problem and for the fully discretization.
Superconvergence of a Galerkin FEM for Higher-Order Elements in Convection-Diffusion Problems
本文研究高阶Galerkin有限元法在奇异摄动对流-扩散问题中的超收敛性，通过解分解和特殊有限元空间表示，证明了能量范数下p+1/4阶的超收敛性质。
Unisolvence of randomized MultiQuadric Kansa collocation for convection-diffusion with mixed boundary conditions
本文证明了在一般区域和混合边界条件下，采用固定配置点与随机虚拟中心的MultiQuadric Kansa方法求解稳态对流扩散方程几乎必然唯一可解，推进了非对称Kansa配置的唯一可解性开放问题。
A finite difference method is constructed to solve singularly perturbed convection-diffusion problems posed on smooth domains. Constraints are imposed on the data so that only regular exponential boundary layers appear in the solution. A domain decomposition method is used, which uses a rectangular grid outside the boundary layer and a Shishkin mesh, aligned to the curvature of the outflow boundary, near the boundary layer. Numerical results are presented to demonstrate the effectiveness of the proposed numerical algorithm.
We propose a new Eulerian-Lagrangian Runge-Kutta finite volume method for numerically solving convection and convection-diffusion equations. Eulerian-Lagrangian and semi-Lagrangian methods have grown in popularity mostly due to their ability to allow large time steps. Our proposed scheme is formulated by integrating the PDE on a space-time region partitioned by approximations of the characteristics determined from the Rankine-Hugoniot jump condition; and then rewriting the time-integral form into a time differential form to allow application of Runge-Kutta (RK) methods via the method-of-lines …
We study the numerical approximation of singularly perturbed convection-diffusion problems on one-dimensional pipe networks. In the vanishing diffusion limit, the number and type of boundary conditions and coupling conditions at network junctions changes, which gives rise to singular layers at the outflow boundaries of the pipes. A hybrid discontinuous Galerkin method is proposed, which provides a natural upwind mechanism for the convection-dominated case. Moreover, the method automatically handles the variable coupling and boundary conditions in the vanishing diffusion limit, leading to an as…
A high order accurate bound-preserving compact finite difference scheme for scalar convection diffusion equations
本文证明经典四阶紧致有限差分格式结合高阶强稳定保持时间离散满足弱单调性，可用简单限制器在不损失守恒性和高阶精度的条件下实现保界性质，并探讨了更高阶格式。
Convergence of adaptive mixed finite element method for convection-diffusion-reaction equations
本文研究了自适应混合有限元法在对流-扩散-反应方程中的收敛性，提出了一种无需准正交性的分析技术，并通过数值实验验证了理论结果。
A Simple Weak Galerkin Finite Element Method for Convection-Diffusion-Reaction Equations on Nonconvex Polytopal Meshes
本文提出了一种适用于非凸多边形网格的简单弱伽辽金有限元方法，用于高效求解对流-扩散-反应方程，并通过实验验证了其理论收敛性和计算效率。
Maximum-norm a posteriori error bounds for an extrapolated upwind scheme applied to a singularly perturbed convection-diffusion problem
本文针对一维奇异扰动对流扩散问题，将Richardson外推法应用于一阶迎风差分格式，推导了任意网格上的鲁棒后验误差界，并证明其能引导自适应网格算法解析层和奇点。
A genuinely stable Lagrange-Galerkin scheme for convection-diffusion problems
本文提出一种无需数值积分的拉格朗日-伽辽金格式，解决对流-扩散问题的稳定性问题，证明了该格式的稳定性和收敛性，数值结果验证了理论分析。
An HDG Method for Dirichlet Boundary Control of Convection Dominated Diffusion PDE
本文提出了一种混合不连续Galerkin方法来近似对流主导的Dirichlet边界控制问题的解，解决了传统方法在处理低正则性和尖锐层时的挑战，并给出了最优的先验误差估计。
Analysis of the SDFEM in a modified streamline diffusion norm for singularly perturbed convection diffusion problems
本文研究了修正流体扩散范数中SDFEM在奇异摄动对流扩散问题中的应用，通过调整稳定化参数获得统一估计，提升数值方案的稳定性与精度。
A finite volume scheme for convection-diffusion equations with nonlinear diffusion derived from the Scharfetter-Gummel scheme
针对非线性扩散的对流扩散方程，提出一种基于Scharfetter-Gummel格式扩展的有限体积格式，该格式保持稳态并证明非退化情形下的收敛性，数值实验验证了其保持长时间行为的高效性。
We consider a system of two singularly perturbed Boundary Value Problems (BVPs) of convection-diffusion type with discontinuous source terms and a small positive parameter multiplying the highest derivatives. Then their solutions exhibit boundary layers as well as weak interior layers. A numerical method based on finite element method (Shishkin and Bakhvalov-Shishkin meshes) is presented. We derive an error estimate of order $O(N^{-1}\ln^{3/2}{N})$ in the energy norm with respect to the perturbation parameter. Numerical experiments are also presented to support our theoretical results.
Adapted Lie splitting method for convection-diffusion problems with singular convective term
针对无界对流项导致对流扩散问题分裂方法失稳的现象，提出了Adapted Lie splitting method，在解析半群框架下证明了一阶收敛性，并通过数值实验验证了其有效性。
We present a Petrov-Gelerkin (PG) method for a class of nonlocal convection-dominated diffusion problems. There are two main ingredients in our approach. First, we define the norm on the test space as induced by the trial space norm, i.e., the optimal test norm, so that the inf-sup condition can be satisfied uniformly independent of the problem. We show the well-posedness of a class of nonlocal convection-dominated diffusion problems under the optimal test norm with general assumptions on the nonlocal diffusion and convection kernels. Second, following the framework of Cohen et al.~(2012), we …
We consider a second order singularly perturbed boundary value problem, of reaction-convection-diffusion type with two small parameters, and the approximation of its solution by the $hp$ version of the Finite Element Method on the so-called {\emph{Spectral Boundary Layer}} mesh. We show that the method converges uniformly, with respect to both singular perturbation parameters, at an exponential rate when the error is measured in the energy norm. Numerical examples are also presented, which illustrate our theoretical findings.
Numerical Methods for Solving Convection-Diffusion Problems
本文讨论了对流扩散问题的空间离散化基本类型（有限差分、有限体积、有限元），研究了离散算子的关键性质，并开发了无条件稳定的显式-隐式时间格式。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。