Eduardo Abreu, Angel Durán
This paper is concerned with the numerical approximation of the Dirichlet initial-boundary-value problem of nonlinear pseudo-parabolic equations with spectral methods. Error estimates for the semidiscrete Galerkin and collocation schemes based on Jacobi polynomials are derived.
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We introduce and analyze a class of Galerkin-collocation discretization schemes in time for the wave equation. Its conceptual basis is the establishment of a direct connection between the Galerkin method for the time discretization and the classical collocation methods, with the perspective of achieving the accuracy of the former with reduced computational costs provided by the latter in terms of less complex linear algebraic systems. Continuously differentiable in time discrete solutions are obtained by the application of a special quadrature rule involving derivatives. Optimal order error es…
We propose and study numerically the implicit approximation in time of the Navier-Stokes equations by a Galerkin-collocation method in time combined with inf-sup stable finite element methods in space. The conceptual basis of the Galerkin-collocation approach is the establishment of a direct connection between the Galerkin method and the classical collocation methods, with the perspective of achieving the accuracy of the former with reduced computational costs in terms of less complex algebraic systems of the latter. Regularity of higher order in time of the discrete solution is ensured furthe…
Optimal Error Estimates for Semidiscrete Galerkin approximations to the Equations of Motion Described by Kelvin-Voigt Viscoelastic Fluid Flow Model
本文研究了凯文-沃伊特粘弹性流体流动模型运动方程的半离散伽辽金方法，推导了精确解的先验估计，并建立了速度和压力在不同范数下的最优误差估计，同时通过数值实验验证了理论结果。
Error estimates for a semidiscrete finite element method for fractional order parabolic equations
针对时间分数阶扩散方程，采用分段线性函数的Galerkin FEM和集中质量Galerkin FEM两种半离散逼近格式，建立了关于解正则性的最优误差估计，包括非光滑初始数据情形。
We consider the periodic initial-value problem for the Serre equations of water-wave theory and its semidiscrete approximation in the space of smooth periodic polynomial splines. We prove that the semidiscrete problem is well posed, locally in time, and satisfies a discrete positivity property for the water depth.
Optimal Dirichlet boundary control for a fractional/normal evolution with a final observation is considered. The unique existence of the solution and the first-order optimality condition of the optimal control problem are derived. The convergence of a temporally semidiscrete approximation is rigorously established, where the control is not explicitly discretized and the state equation is discretized by a discontinuous Galerkin method in time. Numerical results are provided to verify the theoretical results.
Reduced Collocation Methods: Reduced Basis Methods in the Collocation Framework
本文提出了两种适用于配置框架的约化基方法（最小二乘约化配置法和经验约化配置法），在保持高精度和高效率的同时，消除了伽辽金方法中非仿射问题所需的昂贵在线计算。
In this paper, we present and study $C^1$ Petrov-Galerkin and Gauss collocation methods with arbitrary polynomial degree $k$ ($\ge 3$) for one-dimensional elliptic equations. We prove that, the solution and its derivative approximations converge with rate $2k-2$ at all grid points; and the solution approximation is superconvergent at all interior roots of a special Jacobi polynomial of degree $k+1$ in each element, the first-order derivative approximation is superconvergent at all interior $k-2$ Lobatto points, and the second-order derivative approximation is superconvergent at $k-1$ Gauss poi…
Numerical study of Galerkin-collocation approximation in time for the wave equation
本文提出一种基于Galerkin-配位的高阶时空有限元方法，用于波动方程的时间隐式逼近，旨在结合Galerkin精度与配位方法低计算成本，并通过数值实验验证其精度与效率。
A fractional spline collocation-Galerkin method for the time-fractional diffusion equation
本文提出了一种基于分数样条的collocation-Galerkin方法，用于求解具有分数阶时间导数的扩散微分问题，通过闭式表达式简化了分数阶导数的计算，提高了数值方法的准确性和效率。
Optimal-order isogeometric collocation at Galerkin superconvergent points
本文研究了基于最小二乘插值方法和变分插值方法的同构几何插值方法的收敛阶，通过选择伽辽金超收敛点提升收敛性，实现奇数次B-样条/NURBS的最优L²收敛性。
In this paper, discontinuous Galerkin finite element methods are applied to one dimensional Rosenau equation. Theoretical results including consistency, a priori bounds and optimal error estimates are established for both semidiscrete and fully discrete schemes. Numerical experiments are performed to validate the theoretical results. The decay estimates are verified numerically for the Rosenau equation.
In this work, we propose and investigate stable high-order collocation-type discretisations of the discontinuous Galerkin method on equidistant and scattered collocation points. We do so by incorporating the concept of discrete least squares into the discontinuous Galerkin framework. Discrete least squares approximations allow us to construct stable and high-order accurate approximations on arbitrary collocation points, while discrete least squares quadrature rules allow us their stable and exact numerical integration. Both methods are computed efficiently by using bases of discrete orthogonal…
Implicit multiderivative collocation solvers for linear partial differential equations with discontinuous Galerkin spatial discretizations
本文提出了一种新的离散化方法用于不稳对流扩散方程，结合多导数时间积分器和减少未知数数量的新型离散化方法，展示了六阶完全隐式摄动方法的高精度和大时间步长能力。
A collocation scheme that is equivalent to discontinuous Galerkin discretizations
本文证明了一种具有求和-分部性质的谱配置算子产生的半离散化与使用相同底层求积的DG半离散化等价，并研究了等价性条件及数值验证。
Discontinuous Galerkin Method for the Air Pollution Model
本文采用间断伽辽金方法求解二维空气污染模型，证明了半离散常微分方程组解的存在唯一性，并给出了数值误差估计。
Collocation boundary element methods for integral equations are easier to implement than Galerkin methods because the elements of the discretization matrix are given by lower-dimensional integrals. For that same reason, the matrix assembly also requires fewer computations. However, collocation methods typically yield slower convergence rates and less robustness, compared to Galerkin methods. We explore the extent to which oversampled collocation can improve both robustness and convergence rates. We show that in some cases convergence rates can actually be higher than the corresponding Galerkin…
Spectral analysis of matrices in collocation methods based on generalized B-splines with high smoothness
本文研究基于高光滑广义B-样条的同构法中矩阵的谱特性，用于近似具有变量系数和通用几何映射的椭圆PDE。
Stabilized Isogeometric Collocation Methods For Scalar Transport and Incompressible Fluid Flow
本文提出面向标量输运与不可压缩流动的稳定等几何配点格式，借鉴 SUPG/PSPG 思想抑制对流和压力不稳定，并在多类方程上验证其精度与高阶收敛性。
Error Analysis of Semidiscrete Finite Element Methods for Inhomogeneous Time-Fractional Diffusion
本文分析了时间分数扩散方程的半离散有限元方法，针对非光滑右端数据，两种方案均获得几乎最优误差估计，且lumped质量法需对称网格，通过数值实验验证理论结果。
Discrete maximal parabolic regularity for Galerkin finite element methods
本文旨在建立线性抛物方程时间不连续伽辽金解的时间半离散和空间时间全离散最大抛物正则性估计，该估计在非希尔伯特规范中建立最优先验误差估计至关重要。
Discontinuous Galerkin discretization of coupled poroelasticity-elasticity problems
针对孔隙弹性-弹性介质中的波传播问题，提出并分析了多面体网格上的时空有限元间断伽辽金方法，证明了稳定性和误差估计，并通过数值实验验证了理论结果。
A Convergent Hybridizable Discontinuous Galerkin Method for Einstein--Scalar Equations
本文提出并分析了一种混合不连续伽辽金方法，用于球对称的爱因斯坦-标量系统。通过引入缩放变量将模型转化为局部一阶PDE-ODE系统，构建半离散方案，仅对主演化变量进行时间推进，证明了局部半离散良好性，并获得了主演化变量的最优阶误差界。
Semidiscrete Finite Element Analysis of Time Fractional Parabolic Problems: A Unified Approach
本文研究了涉及时间Caputo导数的时分数学抛物问题的数值近似，推导了半离散Galerkin有限元方法的最优误差估计，适用于光滑和非光滑初值条件，并讨论了多项时分数学模型的扩展。
A multi-fidelity stochastic collocation method using locally improved reduced-order models
本文提出一种改进的随机置信法，通过局部改进的降阶模型处理复杂方程，如Burgers方程和纳维-斯托克斯方程，展现高效可靠的特点。
We introduce and analyze a stress-based formulation for Zener's model in linear viscoelasticity. The method is aimed to tackle efficiently heterogeneous materials that admit purely elastic and viscoelastic parts in their composition. We write the mixed variational formulation of the problem in terms of a class of tensorial wave equation and obtain an energy estimate that guaranties the well-posedness of the problem through a standard Galerkin procedure. We propose and analyze mixed continuous and discontinuous Galerkin space discretizations of the problem and derive optimal error bounds for ea…
We present a hybrid numerical-asymptotic (HNA) boundary element method (BEM) for high frequency scattering by two-dimensional screens and apertures, whose computational cost to achieve any prescribed accuracy remains bounded with increasing frequency. Our method is a collocation implementation of the high order hp HNA approximation space of Hewett et al. IMA J. Numer. Anal. 35 (2015), pp.1698- 1728, where a Galerkin implementation was studied. An advantage of the current collocation scheme is that the one-dimensional highly oscillatory singular integrals appearing in the BEM matrix entries are…
A multi-fidelity stochastic collocation method for parabolic PDEs with random input data
本文提出一种结合确定性模型降阶技术的随机插值方法，用于求解带有随机强迫项的抛物型偏微分方程，通过严谨的收敛分析和数值实验证明其可靠性、鲁棒性和高效性。
This paper investigate on numerical analysis on modified Single-layer approach to exterior Dirichlet problem of Laplace equation. We complete the convergence and error analysis of Petrov-Galerkin and Galerkin-Collocation methods with trigonometric basis for the induced modified Symm's integral equation of the first kind on analytic boundary. Besides, utilizing the composite trapezial quadrature formula and trigonometric interpolation to handle the singularity in modified logarithmic kernel, we establish the numerical procedure for implementation. On these numerical examples, we compare the eff…
A merged-cell discontinuous Galerkin method for scalar conservation laws on curved domains
本文提出并分析了一种用于曲域上标量守恒律的合并单元间断伽辽金方法，通过矩形合并处理切割单元，证明了半离散$L^2$耗散和最优误差估计，数值实验验证了接近$k+1$的收敛阶。
This paper focuses on the spectral properties of the mass and stiffness matrices for acoustic wave problems discretized with Isogeometric analysis (IGA) collocation methods in space and Newmark methods in time. Extensive numerical results are reported for the eigenvalues and condition numbers of the acoustic mass and stiffness matrices in the reference square domain with Dirichlet, Neumann and absorbing boundary conditions, varying the polynomial degree $p$, mesh size $h$, regularity $k$, of the IGA discretization and the time step $Δt$ and parameter $β$ of the Newmark method. Results on the s…
Fully discrete Galerkin schemes for the nonlinear and nonlocal Hartree equation
研究连续、半离散和全离散框架下时间依赖Hartree方程的存在唯一性、正则性和逼近性质，为物理应用中Hartree动力学的非线性非局部特征的可控数值计算奠定基础。
Petrov-Galerkin and Spectral Collocation Methods for distributed Order Differential Equations
本文提出两种谱准确方法，用于求解分布式阶分数微分方程，基于分数Sturm-Liouville特征问题，采用非多项式基函数和插值函数，进行离散稳定性分析和数值测试。
Discrete maximal parabolic regularity for Galerkin finite element methods for non-autonomous parabolic problem s
本文研究了非自治抛物问题的Galerkin有限元方法的离散最大抛物正则性，建立了时间半离散和时空全离散估计，并在$ L^p(0,T;L^2(Ω)) $范数下获得最佳近似结果。
High-order implicit Galerkin-Legendre spectral method for the two-dimensional Schrodinger equation
本文提出了一种结合Galerkin-Legendre谱方法和隐式龙格-库塔方法，用于求解带有非齐次Dirichlet边界条件和初始条件的二维薛定谔方程，该方法在L^2范数下具有高阶收敛性。
Convergence of an adaptive collocation method for the stationary parametric diffusion equation with finite-dimensional affine coefficient is shown. The adaptive algorithm relies on a recently introduced residual-based reliable a posteriori error estimator. For the convergence proof, a strategy recently used for a stochastic Galerkin method with an hierarchical error estimator is transferred to the collocation setting. Extensions to other variants of adaptive collocation methods (including the classical one proposed in the paper "Dimension-adaptive tensor-product quadratuture" Computing (2003) …
We investigate the error of the (semidiscrete) Galerkin method applied to a semilinear subdiffusion equation in the presence of a nonsmooth initial data. The diffusion coefficient is allowed to depend on time. It is well-known that in such parabolic problems the spatial error increases when time decreases. The rate of this time-dependency is a function of the fractional parameter and the regularity of the initial condition. We use the energy method to find optimal bounds on the error under weak and natural assumptions on the diffusivity. First, we prove the result for the linear problem and th…
A posteriori error bounds for discontinuous Galerkin methods for quasilinear parabolic problems
针对拟线性抛物问题，利用hp-版本内部惩罚间断Galerkin方法进行逼近，基于椭圆重构框架推导能量范数下的后验误差界。
Notes on error estimates for the standard Galerkin-finite element method for the Shallow Water equations
本文研究了浅水方程的一维初始-边界值问题及对称变体问题，通过标准Galerkin有限元法证明了L²误差估计，并探讨了时间离散化方法的最优阶误差分析。
On an a posteriori error analysis of mixed finite element Galerkin approximations to a second order wave equation
本文基于混合椭圆重构和积分工具，推导了二阶线性双曲方程的后验误差估计，分析了隐式时间离散方案并建立了后验误差估计器。
The conforming finite element Galerkin method is applied to discretise in the spatial direction for a class of strongly nonlinear parabolic problems. Using elliptic projection of the associated linearised stationary problem with Gronwall type result, optimal error estimates are derived, when piecewise polynomials of degree $r\geq 1$ are used, which improve upon earlier results of Axelsson [Numer. Math. 28 (1977), pp. 1-14] requiring for 2d $r\geq 2$ and for 3d $r\geq 3.$ Based on quasi-projection technique introduced by Douglas {\it et al.} [Math. Comp.32 (1978),pp. 345-362], superconvergence …
In recent work (Maierhofer & Huybrechs, 2022, Adv. Comput. Math.), the authors showed that least-squares oversampling can improve the convergence properties of collocation methods for boundary integral equations involving operators of certain pseudo-differential form. The underlying principle is that the discrete method approximates a Bubnov$-$Galerkin method in a suitable sense. In the present work, we extend this analysis to the case when the integral operator is perturbed by a compact operator $\mathcal{K}$ which is continuous as a map on Sobolev spaces on the boundary, $\mathcal{K}:H^{p}\r…
Notes on error estimates for Galerkin approximations of the 'classical' Boussinesq system and related hyperbolic problems
针对一维“经典”Boussinesq系统及其对称形式，采用Galerkin有限元方法进行空间离散，并证明半离散问题和显式Runge-Kutta时间步进全离散问题的误差估计，理论收敛阶与数值实验一致。
The Galerkin Finite Element Method for A Multi-term Time-Fractional Diffusion equation
本文研究了多时间分数导数扩散方程的初始/边界值问题，采用标准Galerkin有限元方法构建空间半离散方案，并推导出初值和非齐次项的近优误差估计，进一步发展了基于时间分数导数有限差分离散的完全离散方案，通过数值实验验证了理论结果的收敛性。
A systematic numerical study on weak Galerkin (WG) finite element method for second order linear parabolic problems is presented by allowing polynomial approximations with various degrees for each local element. Convergence of both semidiscrete and fully discrete WG solutions are established in $L^{\infty}(L^2)$ and $L^{\infty}(H^1)$ norms for a general WG element $({\cal P}_{k}(K),\;{\cal P}_{j}(\partial K),\;\big[{\cal P}_{l}(K)\big]^2)$, where $k\ge 1$, $j\ge 0$ and $l\ge 0$ are arbitrary integers. The fully discrete space-time discretization is based on a first order in time Euler scheme. …
Continuous Galerkin finite element methods for hyperbolic integro-differential equations
针对双曲型积分微分方程，采用连续Galerkin有限元方法，通过能量法和对偶论证证明了最优阶先验误差估计。
Evo-GTransNet for Parabolic PDEs: A Fixed-Feature Galerkin Method of Lines with Quadrature-Mass Orthonormalization
本文提出Evo-GTransNet求解器，通过固定特征Galerkin方法线结合求积质量正交化解决抛物型偏微分方程，实现二阶时间收敛，在基准问题上的平均验证误差优于同类固定特征字典方法。
A numerical method for two-dimensional Boussinesq-Full Dispersion systems modelling internal wave propagation
本文提出一种基于谱Fourier-Galerkin方法和隐式中点规则的数值方法，用于求解二维内波传播的Boussinesq-全频散模型，并分析其收敛性与性能。
Galerkin finite element methods for the Shallow Water equations over variable bottom
本文研究变底地形下的一维浅水方程，提出包含透明边界条件的初边值问题，采用标准Galerkin有限元方法进行空间离散并证明L^2误差估计，结合四阶显式Runge-Kutta时间步进进行数值实验，验证了格式的良好平衡性。
Notes on Galerkin-finite element methods for the Shallow Water equations with characteristic boundary conditions
本文研究了单空间变量下超临界和亚临界浅水方程，采用特征边界条件并利用Galerkin有限元方法进行空间离散，证明了L^2误差估计，并通过数值实验验证了时间离散后方案的优异吸收性能。
A geometry aware arbitrary order collocation Boundary Element Method solver for the potential flow past three dimensional lifting surfaces
本文提出一种基于配点边界元法并结合非线性Kutta条件Galerkin变分公式的三维升力面势流求解器，通过几何感知技术支持任意阶Lagrange单元，在矩形翼和后掠翼上验证了其高精度及重现实验数据的能力。
Linear collective collocation and Galerkin approximations for parametric and stochastic elliptic PDEs
本文研究参数椭圆问题的近似方法，探讨在不同条件下线性集体列点和伽辽金方法的收敛性，证明其在高维参数空间中不产生维度诅咒。
Galerkin FEM for a time-fractional Oldroyd-B fluid problem
本文研究了包含两个 Riemann-Liouville 分数导数的时间分数 Oldroyd-B 流体问题的数值近似，通过建立精确解的正则性结果，分析了基于分段线性 Galerkin 有限元方法的空间半离散方案，并推导了针对光滑和非光滑数据的误差估计，最后通过数值实验验证了方法的有效性。
A priori error estimates of fully discrete finite element Galerkin method for Kelvin-Voigt viscoelastic fluid flow model
本文研究了在forcing function属于$ L^{\infty}(L^2) $时，Kelvin-Voigt粘弹性流体模型的有限元Galerkin方法的先验误差估计，推导了与退化时间$ κ $无关的新的速度和压力界，并讨论了半离散方法在$ L^{\infty} (L^2) $和$ L^{\infty}(H^1_0) $-范数下的最优误差估计，以及初始条件仅在$ H^2\capH_0^1 $时的均匀性。进一步，在唯一性条件下，这些估计在时间趋于无穷时也是均匀的。对于半离散系统的完全离散化，应用了一阶精度的后向欧拉方法，并建立了最优误差估计。最后通过数值实验验证理论结果。本文的结果比之前有限元分析Kelvin-Voigt流体模型的结果更精确，因为本文的误差估计在$ κ ightarrow 0 $时仍保持均匀性。
On error estimates for Galerkin finite element methods for the Camassa-Holm equation
本文研究了Camassa-Holm方程的Galerkin有限元方法的误差估计，采用标准Galerkin有限元方法对周期性初值问题进行空间离散，并证明了半离散近似的最优阶L²误差估计，同时分析了有限区间内系统形式CH方程的初边值问题的收敛性，并通过四阶显式Runge-Kutta方案构建了高精度稳定的全离散方案进行数值实验。
This paper is concerned with the approximation of linear and nonlinearinitial-boundary-value problems of pseudo-parabolic equations with Dirichlet boundary conditions. They are discretized in space by spectral Galerkin and collocation methods based on Legendre and Chebyshev polynomials. The time integration is carried out suitably with robust schemes attending to qualitative features such as stiffness and preservation of strong stability to simulate nonregular problems more correctly. The corresponding semidiscrete and fully discrete schemes are described and the performance of the methods is …
Error estimates for Galerkin approximations of the Serre equations
本文研究了塞雷方程的伽辽金有限元方法在空间离散中的误差估计，采用四阶Runge-Kutta方法构建了高精度离散方案，用于近似孤立波解并研究了初始轮廓分解为孤立波序列及同向不同速孤立波碰撞现象。
Semidiscrete Modeling of Dislocation-Disclination Systems: Finite Element Formulation
本文提出一种基于有限元离散和连续内罚不连续伽辽金框架的数值方法，用于求解含刃型位错与楔形位错偶的有限系统，并通过解析基准验证及多种构型分析，揭示不相容性对应力状态的影响。
Hybridizable discontinuous Galerkin methods for poroelastic wave propagation with symmetric stress approximation
本文提出混合不连续Galerkin方法用于孔隙弹性波方程，结合HDG+方法和LDG-H方法，实现对称应力张量的强对称性近似，并展示近不可压缩材料的鲁棒收敛性。
Optimal Error Estimates of Galerkin Finite Element Methods for Stochastic Partial Differential Equations with Multiplicative Noise
针对带乘性噪声和Lipschitz连续非线性的半线性随机偏微分方程，分析了空间半离散和基于线性隐式Euler-Maruyama方法的时空全离散格式的强收敛误差，得到了最优误差估计。
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