Jin Zhang, Xiaowei Liu
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.
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For a singularly perturbed elliptic model problem with two small parameters, we analyze finite element methods of any order on a Bakhvalov-type mesh. For convergence analysis, we construct a new interpolation by using the characteristics of layers. Besides, a more subtle analysis of the mesh scale near the exponential layer is carried out. Based on the interpolation and new analysis of the mesh scale, we prove the optimal convergence order.
In this paper, the linear finite element method on a Bakhvalov-type mesh is applied to a singularly perturbed problem with two parameters. The solution of the problem exists two exponential boundary layers. A new interpolation, which is simple in construction and analysis, is introduced for convergence analysis. Furthermore, we find a subtle relationship between the Bakhvalov-type mesh itself and the weaker exponential layer and obtain an interesting result. Finally, we prove a supercloseness result between the Lagrange interpolation and the numerical solution. Numerical tests confirm our theo…
Supercloseness of the LDG method for a two-dimensional singularly perturbed convection-diffusion problem on Bakhvalov-type mesh
本文研究二维奇异摄动对流扩散问题，在Bakhvalov型网格上应用LDG方法，通过构建二维局部Gauss-Radau投影和新颖插值，实现了独立于摄动参数的最优阶k+1超逼近。
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。
For singularly perturbed convection-diffusion problems, supercloseness analysis of finite element method is still open on Bakhvalov-type meshes, especially in the case of 2D. The difficulties arise from the width of the mesh in the layer adjacent to the transition point, resulting in a suboptimal estimate for convergence. Existing analysis techniques cannot handle these difficulties well. To fill this gap, a novel interpolation is designed delicately for the first time for the smooth part of the solution, bringing about the optimal supercloseness result of almost order 2 under an energy norm f…
Uniform convergence of finite element method on Bakhvalov-type mesh for a 2-D singularly perturbed convection-diffusion problem with exponential layers
针对二维含指数层奇异扰动对流-扩散问题，提出新型边界插值分析技术与简单插值，证明 Bakhvalov 型网格上任意阶有限元在能量范数下 k+1 阶一致收敛。
Uniform convergence on a Bakhvalov-type mesh using the preconditioning approach: Technical report
本文研究了一维线性奇异摄动对流扩散问题，在Bakhvalov型网格上的离散化分析，并利用预条件技术实现扰动参数下的点wise均匀收敛。
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…
We propose a new analysis of convergence for a $k$th order ($k\ge 1$) finite element method, which is applied on Bakhvalov-type meshes to a singularly perturbed two-point boundary value problem. A novel interpolant is introduced, which has a simple structure and is easy to generalize. By means of this interpolant, we prove an optimal order of uniform convergence with respect to the perturbation parameter. Numerical experiments illustrate these theoretical results.
A two-parameter singularly perturbed problem with discontinuous source and convection coefficient is considered in one dimension. Both convection coefficient and source term are discontinuous at a point in the domain. The presence of perturbation parameters results in boundary layers at the boundaries. Also, an interior layer occurs due to the discontinuity of data at an interior point. An upwind scheme on an appropriately defined Shishkin-Bakhvalov mesh is used to resolve the boundary layers and interior layers. A three-point formula is used at the point of discontinuity. The proposed method …
An efficient spline-based scheme on Shishkin-type meshes for solving singularly perturbed coupled systems with Robin boundary conditions
针对带Robin边界条件、解具重叠相互作用边界层的弱耦合奇摄动线性反应扩散方程组，提出基于Shishkin型网格的三次样条结合中心差分的高效格式，实现近二阶及精确二阶收敛，并用数值结果验证。
In this article, we have considered a time-dependent two-parameter singularly perturbed parabolic problem with discontinuous convection coefficient and source term. The problem contains the parameters $ε$ and $μ$ multiplying the diffusion and convection coefficients, respectively. A boundary layer develops on both sides of the boundaries as a result of these parameters. An interior layer forms near the point of discontinuity due to the discontinuity in the convection and source term. The width of the interior and boundary layers depends on the ratio of the perturbation parameters. We discuss t…
In convergence analysis of finite element methods for singularly perturbed reaction--diffusion problems, balanced norms have been successfully introduced to replace standard energy norms so that layers can be captured. In this article, we focus on the convergence analysis in a balanced norm on Bakhvalov-type rectangular meshes. In order to achieve our goal, a novel interpolation operator, which consists of a local weighted $L^2$ projection operator and the Lagrange interpolation operator, is introduced for a convergence analysis of optimal order in the balanced norm. The analysis also depends …
A class of difference schemes uniformly convergent on a modified Bakhvalov mesh
本文研究了一种奇异扰动一维半线性反应扩散问题的数值解法，构造了一类差分方案，并证明了其解的唯一性，主要贡献是证明了在修改后的Bakhvalov网格上，该方案具有二阶$\mathcal{O}(1/N^2)$的$\varepsilon$-均匀收敛性。
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.
A singularly perturbed convection-diffusion problem posed on the unit square in $\mathbb{R}^2$, whose solution has exponential boundary layers, is solved numerically using the local discontinuous Galerkin (LDG) method with piecewise polynomials of degree at most $k>0$ on three families of layer-adapted meshes: Shishkin-type, Bakhvalov-Shishkin-type and Bakhvalov-type.On Shishkin-type meshes this method is known to be no greater than $O(N^{-(k+1/2)})$ accurate in the energy norm induced by the bilinear form of the weak formulation, where $N$ mesh intervals are used in each coordinate direction.…
A class of different schemes for the numerical solving of semilinear singularly--perturbed reaction--diffusion boundary--value problems was constructed. The stability of the difference schemes was proved, and the existence and uniqueness of a numerical solution were shown. After that, the uniform convergence with respect to a perturbation parameter $\varepsilon$ on a modified Shishkin mesh of order 2 has been proven. For such a discrete solution, a global solution based on a linear spline was constructed, also the error of this solution is in expected boundaries. Numerical experiments at the e…
We analyse the local discontinuous Galerkin (LDG) method for two-dimensional singularly perturbed reaction-diffusion problems. A class of layer-adapted meshes, including Shishkin- and Bakhvalov-type meshes, is discussed within a general framework. Local projections and their approximation properties on anisotropic meshes are used to derive error estimates for energy and "balanced" norms. Here, the energy norm is naturally derived from the bilinear form of LDG formulation and the "balanced" norm is artifically introduced to capture the boundary layer contribution. We establish a uniform converg…
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.
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.
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。
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.
A singularly perturbed parabolic problem of convection-diffusion type with a discontinuous initial condition is examined. An analytic function is identified which matches the discontinuity in the initial condition and also satisfies the homogenous parabolic differential equation associated with the problem. The difference between this analytical function and the solution of the parabolic problem is approximated numerically, using an upwind finite difference operator combined with an appropriate layer-adapted mesh. The numerical method is shown to be parameter-uniform. Numerical results are pre…
Fitted finite element methods for singularly perturbed elliptic problems of convection-diffusion type
该研究结合指数样条基函数与Shishkin网格，为二维奇异摄动对流-扩散问题构造满足离散极大值原理的参数一致拟合有限元方法，并证明经典情形二阶、任意参数下一阶的最大范数收敛。
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…
A singularly perturbed parabolic problem of convection-diffusion type with a discontinuous initial condition is examined. A particular complimentary error function is identified which matches the discontinuity in the initial condition. The difference between this analytical function and the solution of the parabolic problem is approximated numerically. A co-ordinate transformation is used so that a layer-adapted mesh can be aligned to the interior layer present in the solution. Numerical analysis is presented for the associated numerical method, which establishes that the numerical method is a…
A singularly perturbed convection-diffusion problem posed on an annulus
本文提出一种有限差分方法，用于解决在环形区域上的一致扰动对流-扩散问题，通过极坐标、上游有限差分算子和分段均匀Shishkin网格结合，确保不形成内部层，并建立参数均匀误差界。
Supercloseness of the HDG method on Shishkin mesh for a singularly perturbed convection diffusion problem in 2D
本文首次分析了Shishkin网格上HDG方法求解二维奇异扰动对流扩散问题的参数一致收敛性，通过新误差控制技术克服层内对流项估计难点，实现近$k+\ rac{1}{2}$阶超接近性。
A parameter uniform fitted mesh method for a weakly coupled system of two singularly perturbed convection-diffusion equations
本文提出了一种参数均匀拟合网格方法，用于求解弱耦合的两个奇异摄动对流-扩散方程系统的边值问题，该方法在最大范数下对扰动参数具有统一的一阶收敛性。
As a popular stabilization technique, the nonsymmetric interior penalty Galerkin (NIPG) method has significant application value in computational fluid dynamics. In this paper, we study the NIPG method for a typical two-dimensional singularly perturbed convection diffusion problem on a Shishkin mesh. According to the characteristics of the solution, the mesh and numerical scheme, a new composite interpolation is introduced. In fact, this interpolation is composed of a vertices-edges-element interpolation within the layer and a local $L^{2}$-projection outside the layer. On the basis of that, b…
A singularly perturbed parabolic problem of convection-diffusion type with incompatible inflow boundary and initial conditions is examined. In the case of constant coefficients, a set of singular functions are identified which match certain incompatibilities in the data and also satisfy the associated homogenous differential equation. When the convective coefficient only depends on the time variable and the initial/boundary data is discontinuous, then a mixed analytical/numerical approach is taken. In the case of variable coefficients and the zero level of compatibility being satisfied (i.e. c…
Multigrid preconditioning of singularly perturbed convection-diffusion equations
本文针对奇异扰动对流扩散方程的有限元离散化线性系统，提出并验证了基于张量积和半结构化网格的高效多重网格迭代求解方法，证明其是标准流动基准的鲁棒预处理子。
Two-grid algorithms for singularly perturbed reaction-diffusion problems on layer adapted meshes
本文提出基于Bellman-Kalaba准线性化和Axelsson-Xu有限元双网格方法的新双网格算法，用于求解退化反应扩散方程，通过粗网格求解廉价问题，细网格求解准线性化后的线性问题，所有方案对小参数均匀收敛。
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。
In this paper, we analyze the supercloseness result of nonsymmetric interior penalty Galerkin (NIPG) method on Shishkin mesh for a singularly perturbed convection diffusion problem. According to the characteristics of the solution and the scheme, a new analysis is proposed. More specifically, Gauß Lobatto interpolation and Gauß Radau interpolation are introduced inside and outside the layer, respectively. By selecting special penalty parameters at different mesh points, we further establish supercloseness of almost k + 1 order under the energy norm. Here k is the order of piecewise polynomials…
Supercloseness of the DDG method for a singularly perturbed convection diffusion problem on Shishkin mesh
本文研究了Shishkin网格上奇异扰动对流扩散问题的DDG方法超接近性，通过设计新的复合插值并选择合适的数值通量参数，在能量范数下获得了几乎$k+1$阶的超接近性结果。
Robust numerical schemes for singularly perturbed delay parabolic convection diffusion problems with degenerate coefficient
本文研究了具有退化系数的奇异 perturbed 时间延迟对流扩散方程的 Dirichlet 边界值问题，通过建立先验的显式界，分析了解及其导数的渐近行为，并提出了一种结合后向欧拉方案和中点迎风与中心差分方案的数值方法，以证明其有效性。
An Upwind Finite Difference Method to Singularly Perturbed Convection Diffusion Problems on a Shishkin Mesh
针对带小参数ε的奇异摄动对流扩散Dirichlet边值问题，提出结合Shishkin网格的迎风格式有限差分法，求解器直接高效、时间随网格点线性缩放，数值验证表明其精度高且代码公开。
A finite difference scheme for two-dimensional singularly perturbed convection-diffusion problem with discontinuous source term
针对带不连续源项的二维奇异扰动对流扩散问题，提出一种在分段均匀网格上适应不连续性的有限差分格式，证明了扰动参数无关的近一阶收敛性，并用数值结果验证了其尖锐性。
Layer-adapted meshes for singularly perturbed problems via mesh partial differential equations and a posteriori information
本文提出一种基于 MPDE 并利用后验解估计构造 SPDE 层适应网格的新方法，给出高效实现算法、一/二维双参数问题数值结果及 FEniCS 代码。
Deep learning-based numerical schemes such as Physically Informed Neural Networks (PINNs) have recently emerged as an alternative to classical numerical schemes for solving Partial Differential Equations (PDEs). They are very appealing at first sight because implementing vanilla versions of PINNs based on strong residual forms is easy, and neural networks offer very high approximation capabilities. However, when the PDE solutions are low regular, an expert insight is required to build deep learning formulations that do not incur in variational crimes. Optimization solvers are also significantl…
Almost nothing is known about the layer structure of solutions to singularly perturbed Caputo fractional boundary value problems. We discuss simple convection-diffusion and reaction-diffusion problems.
A Mixed Finite Element Method for Singularly Perturbed Fourth Oder Convection-Reaction-Diffusion Problems on Shishkin Mesh
针对带小正参数的奇异摄动四阶常微分方程Lidstone边值问题，提出将其分解为二阶系统、结合Shishkin网格的混合有限元直接求解法，精度高且计算时间随网格点线性增长，数值结果验证了理论与方法准确性。
Quasi-Monte Carlo finite element approximation for singularly perturbed convection-diffusion problems with random velocity
针对随机速度场驱动的奇异摄动对流扩散问题，提出结合有限元离散、截断Karhunen–Loève展开和格点拟蒙特卡罗方法的数值框架，证明均方误差界和与维数无关的近线性最优收敛率，且收敛率不依赖于奇异摄动参数。
Expansion of a singularly perturbed equation with a two-scale converging convection term
本文基于双尺度收敛工具，提出一种模型层次结构，用于对任意ε值的分布函数建模，通过递归识别展开项，应用于磁约束聚变和带电粒子束演化中的线性Vlasov方程。
Full analysis of the Green's function for a singularly perturbed convection-diffusion problem in three dimensions
针对三维具有特征层的线性奇异摄动对流扩散问题，建立了格林函数及其导数的L1范数尖锐界，并显式给出了对小摄动参数的依赖关系，为后续数值分析提供基础。
A singularly perturbed convection-diffusion problem,posed on the unit square in $\mathbb{R}^2$, is studied; its solution has both exponential and characteristic boundary layers. The problem is solved numerically using the local discontinuous Galerkin (LDG) method on Shishkin meshes. Using tensor-product piecewise polynomials of degree at most $k>0$ in each variable, the error between the LDG solution and the true solution is proved to converge, uniformly in the singular perturbation parameter, at a rate of $O((N^{-1}\ln N)^{k+1/2})$ in an associated energy norm, where $N$ is the number of mesh…
Wavelet-based Edge Multiscale Finite Element Methods for Singularly Perturbed Convection-Diffusion Equations
提出一种基于小波的边缘多尺度有限元方法（WEMsFEM），通过局部泡函数与调和延拓分裂及单位分解，在网格限制小的条件下高效求解奇异摄动对流扩散方程，并给出经二维和三维数值试验验证的收敛率分析。
Motivated by a wide range of real-world problems whose solutions exhibit boundary and interior layers, the numerical analysis of discretizations of singularly perturbed differential equations is an established sub-discipline within the study of the numerical approximation of solutions to differential equations. Consequently, much is known about how to accurately and stably discretize such equations on \textit{a priori} adapted meshes, in order to properly resolve the layer structure present in their continuum solutions. However, despite being a key step in the numerical simulation process, muc…
This paper presents an extended version of the article [Franz, S., Kopteva, N.: J. Differential Equations, 252 (2012)]. The main improvement compared to the latter is in that here we additionally estimate the mixed second-order derivative of the Green's function. The case of Neumann conditions along the characteristic boundaries is also addressed. A singularly perturbed convection-diffusion problem is posed in the unit square with a horizontal convective direction. Its solutions exhibit parabolic and exponential boundary layers. Sharp estimates of the Green's function and its first- and second…
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.
We present the analysis for an $hp$ weak Galerkin-FEM for singularly perturbed reaction-convection-diffusion problems in one-dimension. Under the analyticity of the data assumption, we establish robust exponential convergence, when the error is measured in the energy norm, as the degree $p$ of the approximating polynomials is increased. The Spectral Boundary Layer mesh is used, which is the minimal (layer adapted) mesh for such problems. Numerical examples illustrating the theory are also presented.
Isogeometric Analysis for singularly perturbed problems in 1-D: a numerical study
本文使用基于B样条的等几何分析对一维反应-对流-扩散型奇异摄动问题进行数值研究，探讨了节点选择策略，以在最大范数下实现一致的指数收敛，并为三种不同问题提供了具体指导方针。
A hybrid approach for singularly perturbed parabolic problem with discontinuous data
本文针对源项和对流系数不连续的二维奇异摄动抛物问题，提出结合中心差分与中点迎风的混合单调差分格式，并证明其空间近二阶、时间二阶的一致收敛性。
A higher order numerical scheme for singularly perturbed parabolic turning point problems exhibiting twin boundary layers
本文提出了一种参数均匀的数值方法，用于求解一维具有双指数边界层的奇异摄动抛物型对流扩散转折点问题。通过分析解及其偏导数的渐近行为，采用隐式欧拉法在均匀网格上进行时间离散，结合混合方案在广义Shishkin网格上进行空间离散，证明了该方案在时间方向上具有ε-均匀收敛性一级，在空间方向上具有二级收敛性（至多一个对数因子）。数值实验验证了理论结果，并与均匀网格和标准Shishkin网格上的迎风方案进行比较，以展示在广义Shishkin网格上所提方案的高阶准确性。
Singular Layer Physics-Informed Neural Network Method for Convection-Dominated Boundary Layer Problems in Two Dimensions
针对二维对流占优奇异摄动边界层问题传统神经网络方法精度不足的缺陷，提出结合奇异层校正函数的半解析PINNs方法，显著提升了求解的准确性与稳定性。
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.
A parameter uniform hybrid approach for singularly perturbed two-parameter parabolic problem with discontinuous data
本文针对二维奇异摄动双参数抛物问题中的不连续数据与内部层、边界层，提出空间方向基于Shishkin网格的混合单调差分格式，并结合时间方向Crank-Nicolson方法，实现空间近二阶、时间二阶的一致收敛，数值实验验证了其优越性。
Although Physics-Informed Neural Networks (PINNs) have been successfully applied in a wide variety of science and engineering fields, they can fail to accurately predict the underlying solution in slightly challenging convection-diffusion-reaction problems. In this paper, we investigate the reason of this failure from a domain distribution perspective, and identify that learning multi-scale fields simultaneously makes the network unable to advance its training and easily get stuck in poor local minima. We show that the widespread experience of sampling more collocation points in high-loss laye…
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。
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