Waixiang Cao, Chunmei Wang
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.
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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.
A New Primal-Dual Weak Galerkin Finite Element Method for Ill-posed Elliptic Cauchy Problems
本文提出了一种新的对偶弱伽辽金有限元法，用于求解不良好椭圆柯西问题，该方法具有对称、良好定义且满足精确解的特性，建立了最优阶误差估计和弱L²拓扑下的收敛性，通过数值实验验证了方法的效率和精度。
Primal-Dual Weak Galerkin Finite Element Methods for Elliptic Cauchy Problems
本文提出并分析了一种求解一类病态椭圆柯西问题的数值方案，结合约束最小化方法与弱格雷戈尔有限元方法，构建了对称、良定的系统，给出了离散Sobolev范数下的稳定性及误差估计，并通过数值结果验证理论。
In this article, the authors present a new $L^p$- primal-dual weak Galerkin method ($L^p$-PDWG) for convection-diffusion equations with $p>1$. The existence and uniqueness of the numerical solution is discussed, and an optimal-order error estimate is derived in the $L^q$-norm for the primal variable, where $\frac 1p+\frac 1q=1$. Furthermore, error estimates are established for the numerical approximation of the dual variable in the standard $W^{m,p}$ norm, $0\le m\le 2$. Numerical results are presented to demonstrate the efficiency and accuracy of the proposed $L^p$-PDWG method.
A Primal Staggered Discontinuous Galerkin Method on Polytopal Meshes
针对多面体网格上的椭圆方程求解问题，提出一种采用原始-对偶网格框架的新型可杂交交错间断Galerkin方法，兼具通量局部守恒、自由度低、最优收敛与高效的优势，并建立了与其他数值方法的关联。
Parallelization of continuous and discontinuous Galerkin dual-primal Isogeometric tearing and interconnecting methods
本文研究了用于求解大规模连续与间断伽辽金方程组的双优先等几何撕裂与连接方法的并行化，通过能量最小化主子空间实现，并展示了二维和三维问题的弱标度和强标度研究结果。
A new primal-dual weak Galerkin (PD-WG) finite element method was developed and analyzed in this article for first-order linear convection equations in non-divergence form. The PD-WG method results in a symmetric discrete system involving not only the original equation for the primal variable, but also the dual/adjoint equation for the dual variable (also known as Lagrangian multiplier). Optimal order of error estimates in various discrete Sobolev norms are derived for the numerical solutions arising from the PD-WG scheme. Numerical results are produced and reported to illustrate the accuracy …
Low Regularity Primal-Dual Weak Galerkin Finite Element Methods for Convection-Diffusion Equations
本文提出了一种基于原始-对偶弱Galerkin（PDWG）有限元框架的数值方法，用于求解低正则性假设下的对流扩散问题，证明了其稳定性和收敛性，并给出了原始变量的先验误差估计与数值验证。
A Parallel iterative Algorithm for primal-dual weak Galerkin Schemes
本文针对求解泊松方程的原始-对偶弱伽辽金有限元方法，提出一种基于区域分解的并行迭代算法，证明了解的存在唯一性，并获得了离散范数和$L^2$范数下的最优阶误差估计。
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…
A Primal-Dual Weak Galerkin Finite Element Method for Fokker-Planck Type Equations
本文提出了一种用于福克-计划克类型方程第二类椭圆方程的对偶原弱有限元方法，通过变分形式实现无正则性要求的数值解，证明了最优收敛性并展示了数值结果。
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.
This paper presents a new $L^p$-primal-dual weak Galerkin (PDWG) finite element method for the div-curl system with the normal boundary condition for $p>1$. Two crucial features for the proposed $L^p$-PDWG finite element scheme are as follows: (1) it offers an accurate and reliable numerical solution to the div-curl system under the low $W^{α, p}$-regularity ($α>0$) assumption for the exact solution; (2) it offers an effective approximation of the normal harmonic vector fields on domains with complex topology. An optimal order error estimate is established in the $L^q$-norm for the primal vari…
A simplified primal-dual weak Galerkin (S-PDWG) finite element method is designed for the Fokker-Planck type equation with non-smooth diffusion tensor and drift vector. The discrete system resulting from S-PDWG method has significantly fewer degrees of freedom compared with the one resulting from the PDWG method proposed by Wang-Wang \cite{WW-fp-2018}. Furthermore, the condition number of the S-PDWG method is smaller than the PDWG method \cite{WW-fp-2018} due to the introduction of a new stabilizer, which provides a potential for designing fast algorithms. Optimal order error estimates for the…
A primal discontinuous Galerkin method with static condensation on very general meshes
本文提出了一种高效的原始不连续伽辽金方法变种，用于求解二阶椭圆方程，该方法在非常一般的网格（多面体，可能带有曲面边界）上使用内部惩罚项。通过静默消元（即逐元素消除某些自由度）实现效率，特别是在使用高阶多项式时。这种方法在保持最优误差估计的同时改变了原始方法。数值实验表明，新方法产生的解与经典方法非常接近。
A Primal-Dual Weak Galerkin Method for Div-Curl Systems with low-regularity solutions
本文提出一种求解具有切向边界条件和低正则性解的 div-curl 系统的原对偶弱 Galerkin 有限元方法，基于无偏导数的弱变分形式及对偶问题，在 $L^2$ 中建立最优阶误差估计，并通过数值结果验证理论收敛性。
A Primal-Dual Weak Galerkin Finite Element Method for Second Order Elliptic Equations in Non-Divergence Form
本文提出了一种新的数值算法，用于求解非散度形式的二阶椭圆方程。该方法基于弱伽辽金策略局部构造离散弱Hessian算子，通过最小化非负二次泛函并引入拉格朗日乘子变量，得到对称有限元方案，并推导了最优误差估计。
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。
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。
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。
Analysis of discontinuous Galerkin dual-primal isogeometric tearing and interconnecting methods
本文分析了在仅考虑顶点原始变量的二维情况下，不连续伽辽金双优先等几何撕裂和连接方法（dG-IETI-DP）的性能，推导了预处理系统条件数的近似最优界。
A modified primal-dual weak Galerkin (M-PDWG) finite element method is designed for the second order elliptic equation in non-divergence form. Compared with the existing PDWG methods proposed in \cite{wwnondiv}, the system of equations resulting from the M-PDWG scheme could be equivalently simplified into one equation involving only the primal variable by eliminating the dual variable (Lagrange multiplier). The resulting simplified system thus has significantly fewer degrees of freedom than the one resulting from existing PDWG scheme. In addition, the condition number of the simplified system …
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…
Dual-Primal Isogeometric Tearing and Interconnecting Solvers for large-scale systems of multipatch continuous Galerkin IgA equations
本文提出双模双原等几何撕裂与连接求解器，用于解决大规模多补丁连续伽辽金IgA方程系统，针对多补丁域的条件数估计和C^0光滑性下的预处理系统进行理论分析与数值验证。
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。
Local Discontinuous Galerkin Methods for Solving Convection-Diffusion and Cahn-Hilliard Equations on Surfaces
本文提出了求解静态2D流形上二阶和四阶时间依赖偏微分方程的局部间断Galerkin方法，该方法通过平面三角形剖分和数值通量设计实现二阶精度和能量稳定，并通过数值实验验证了其有效性。
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。
We investigate numerical behaviour of a convection diffusion equation with random coefficients by approximating statistical moments of the solution. Stochastic Galerkin approach, turning the original stochastic problem to a system of deterministic convection diffusion equations, is used to handle the stochastic domain in this study, whereas discontinuous Galerkin method is used to discretize spatial domain due to its local mass conservativity. A priori error estimates of the stationary problem and stability estimate of the unsteady model problem are derived in the energy norm. To address the c…
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 …
This paper investigates superconvergence properties of the local discontinuous Galerkin methods with generalized alternating fluxes for one-dimensional linear convection-diffusion equations. By the technique of constructing some special correction functions, we prove the $(2k+1)$th order superconvergence for the cell averages, and the numerical traces in the discrete $L^2$ norm. In addition, superconvergence of order $k+2$ and $k+1$ are obtained for the error and its derivative at generalized Radau points. All theoretical findings are confirmed by numerical experiments.
Convergence analysis of a weak Galerkin finite element method on a Bakhvalov-type mesh for a singularly perturbed convection-diffusion equation in 2D
本文提出二维奇异摄动对流扩散方程在Bakhvalov型网格上的弱Galerkin有限元方法，推导误差估计并证明最优收敛阶，数值实验验证了方法的有效性。
We investigate a numerical behaviour of robust deterministic optimal control problem subject to a convection diffusion equation containing uncertain inputs. Stochastic Galerkin approach, turning the original optimization problem containing uncertainties into a large system of deterministic problems, is applied to discretize the stochastic domain, while a discontinuous Galerkin method is preferred for the spatial discretization due to its better convergence behaviour for optimization problems governed by convection dominated PDEs. Error analysis is done for the state and adjoint variables in th…
A discontinuous Galerkin multiscale method for convection-diffusion problems
本文提出一种不连续伽辽金局部正交分解多尺度方法，用于处理粗糙、非均匀且高变系数的对流-扩散问题，通过局部化基底捕捉多尺度特征和内部/边界层。
Discontinuous Galerkin method for fractional convection-diffusion equations
本文提出了一种用于分数输运-扩散方程的不连续Galerkin方法，通过分数拉普拉斯算子定义分数阶次α(1<α<2)，并证明了子扩散问题的稳定性及最优收敛性，以及一般分数输运-扩散问题的收敛性。
Convergence analysis of novel discontinuous Galerkin methods for a convection dominated problem
本文针对对流占优的对流-扩散-反应方程，基于新型 DG 微分学框架，结合 DWDG 与平均离散梯度算子构造稳定数值格式，并证明其最优收敛性。
This paper is concerned with superconvergence properties of the direct discontinuous Galerkin (DDG) method for two-dimensional nonlinear convection-diffusion equations. By using the idea of correction function, we prove that, for any piecewise tensor-product polynomials of degree $k\geq 2$, the DDG solution is superconvergent at nodes and Lobatto points, with an order of ${\cal O}(h^{2k})$ and ${\cal O}(h^{k+2})$, respectively. Moreover, superconvergence properties for the derivative approximation are also studied and the superconvergence points are identified at Gauss points, with an order of…
The local discontinuous Galerkin (LDG) method is studied for a third-order singularly perturbed problem of the convection-diffusion type. Based on a regularity assumption for the exact solution, we prove almost $O(N^{-(k+1/2)})$ (up to a logarithmic factor) energy-norm convergence uniformly in the perturbation parameter. Here, $k\geq 0$ is the maximum degree of piecewise polynomials used in discrete space, and $N$ is the number of mesh elements. The results are valid for the three types of layer-adapted meshes: Shishkin-type, Bakhvalov-Shishkin type, and Bakhvalov-type. Numerical experiments a…
Multigrid preconditioning for discontinuous Galerkin discretizations of an elliptic optimal control problem with a convection-dominated state equation
本文研究对流占优约束下椭圆分布最优控制问题的间断 Galerkin 离散，证明显式含参数的最优收敛率，并提出仅需两次多重网格近似求解及块 Gauss-Seidel 扫描的下风多层预处理器。
We introduce a local adaptive discontinuous Galerkin method for convection-diffusion-reaction equations. The proposed method is based on a coarse grid and iteratively improves the solution's accuracy by solving local elliptic problems in refined subdomains. For purely diffusion problems, we already proved that this scheme converges under minimal regularity assumptions [A. Abdulle and G.Rosilho de Souza, ESAIM: M2AN, 53(4):1269--1303, 2019]. In this paper, we provide an algorithm for the automatic identification of the local elliptic problems' subdomains employing a flux reconstruction strategy…
A weak Galerkin least squares finite element method for linear convection equations in non-divergence form
本文提出了一种适用于非散度形式线性对流方程的弱伽辽金最小二乘有限元方法，无需强制性假设，具备最优阶误差估计和高精度高效性。
Hybridized discontinuous Galerkin method for convection-diffusion problems
本文提出一种新的混合不连续伽辽金方法，用于求解具有混合边界条件的对流-扩散问题，通过减少全局耦合自由度提升效率，并证明了在粘性系数小时解的近似性。
A Locking-Free Weak Galerkin Finite Element Method for Elasticity Problems in the Primal Formulation
本文提出了一种任意阶次非锁态数值方案，用于线弹性问题在一般多边形/多面体划分上，采用弱伽辽金有限元方法。通过在每个元素上求解本地问题来引入离散弱应变和应力张量，从而获得最优阶次的非锁态误差估计。
Discontinuous Galerkin methods and their adaptivity for the tempered fractional (convection) diffusion equations
本文研究了不连续伽辽金方法在 tempered 分数扩散方程中的自适应性，通过内罚项和数值通量处理扩散和对流项，提供了稳定性与收敛性分析，并通过数值实验验证了自适应方法的有效性。
In this paper, we present optimal error estimates of the local discontinuous Galerkin method with generalized numerical fluxes for one-dimensional nonlinear convection-diffusion systems. The upwind-biased flux with adjustable numerical viscosity for the convective term is chosen based on the local characteristic decomposition, which is helpful in resolving discontinuities of degenerate parabolic equations without enforcing any limiting procedure. For the diffusive term, a pair of generalized alternating fluxes are considered. By constructing and analyzing generalized Gauss-Radau projections wi…
Preconditioned discontinuous Galerkin method and convection-diffusion-reaction problems with guaranteed bounds to resulting spectra
本文提出一种适用于Galerkin与间断Galerkin离散的二阶偏微分方程预条件方法，可为对称问题全部特征值及非对称问题特征值实虚部提供保证界，并通过数值实验验证有效性。
An a posteriori error estimator for discontinuous Galerkin methods for non-stationary convection-diffusion problems
针对线性非定常对流扩散初边值问题的间断Galerkin方法，提出了一种鲁棒的后验误差估计子，并实现了自适应算法，数值实验验证了估计子对Péclet数的鲁棒性以及自适应算法的最优收敛阶。
Stability of an upwind Petrov Galerkin discretization of convection diffusion equations
本文研究了小粘性条件下对流扩散方程的数值方法，提出一种指数拟合的 conforming Petrov-Galerkin 方法，并证明了在特定条件下该方法的稳定性和连续性。
An posteriori error estimator for discontinuous Galerkin discretisations of convection-diffusion problems with application to Earth's mantle convection simulations
本文针对强对流主导的对流-扩散问题提出基于指数拟合和椭圆重构的新后验误差估计，降低了对Gronwall型指数的依赖，并在地球地幔对流模拟中结合ASPECT验证了其在网格自适应中的优越效果。
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…
A posteriori error estimator based on gradient recovery by averaging for convection-diffusion-reaction problems approximated by discontinuous Galerkin methods
针对间断Galerkin方法逼近的对流-扩散-反应问题，提出两种基于平均梯度恢复的后验误差估计子，并证明其可靠性和效率，数值实验验证了理论结果。
Third order Maximum-Principle-Satisfying Direct discontinuous Galerkin methods for time dependent convection diffusion equations on unstructured triangle mesh
本文提出了一种三阶最大原理满足的直接不连续伽辽金方法，用于在无结构三角网格上求解时间依赖的对流扩散方程，通过合理选择参数对(β₀,β₁)确保二次多项式解满足严格最大原理，保持第三阶精度。
On convergence of upwinding Petrov-Galerkin methods for convection-diffusion
针对对流扩散问题，研究迎风Petrov-Galerkin离散格式，推导一维情形离散矩阵逆公式，提出带缩放参数的二次泡迎风方法并证明其最优收敛性，推广到二维问题，为多维对流占优模型提供新离散思路。
A Least-Squares Weak Galerkin Finite Element Scheme for Cauchy Problems in Convection--Diffusion
本文提出了一种最小二乘弱伽辽金有限元方法，用于求解严重病态的对流-扩散方程Cauchy问题，通过弱导数处理复杂边界条件和内部界面，建立对称正定离散线性系统，并证明了解的唯一性和最优误差估计。
In this paper we analyze the error as well for the semi-discretization as the full discretization of a time-dependent convection-diffusion problem. We use for the discretization in space the local discontinuous Galerkin (LDG) method on a class of layer-adapted meshes including Shishkin-type and Bakhvalov-type meshes and the implicit $θ$-scheme in time. For piecewise tensor-product polynomials of degree $k$ we obtain uniform or almost uniform error estimates with respect to space of order $k+1/2$ in some energy norm and optimal error estimates with respect to time. Our analysis is based on care…
Discontinuous Galerkin method based on the reduced space for the nonlinear convection-diffusion-reaction equation
针对非线性对流-扩散-反应方程，提出基于勒让德矩重构算子的约化DG空间，结合隐显时间推进LDG方法，在保证精度的同时减少自由度提升效率，并完成理论分析与数值验证。
A hybridized discontinuous Galerkin method for 2D fractional convection-diffusion equations
本文提出了一种混合间断伽辽金方法，用于求解包含空间分数阶导数的二维分数对流-扩散方程，通过特征法和混合间断伽辽金方法构建对称变分公式，证明了方法的稳定性，并通过数值实验验证了其收敛性。
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。
Error estimates of stable and stabilized Lagrange-Galerkin schemes for natural convection problems
本文研究了自然对流问题中稳定和稳定化拉格朗日-伽辽金方案的最优误差估计，采用温和的时间步长和网格大小条件，通过扩展先前文献中纳维-斯托克斯方程误差估计的证明方法进行证明。
An Adaptive Algorithm Based on Stochastic Discontinuous Galerkin for Convection Dominated Equations with Random Data
本文提出一种结合网格细化与多项式阶数自适应的随机间断伽辽金方法，用于求解随机数据对流占优方程，并推导了可靠的残差型误差估计器，通过多个基准算例验证了其有效性。
Local discontinuous Galerkin method for distributed-order time and space-fractional convection-diffusion and Schrödinger type equations
本文提出局部间断伽辽金方法用于求解分布式阶时间与空间分数阶对流扩散和薛定谔类型方程，证明了稳定性及最优收敛阶，通过数值实验验证了收敛阶。
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