Yuwen Li, Ludmil Zikatanov
We present a framework that relates preconditioning with a posteriori error estimates in finite element methods. In particular, we use standard tools in subspace correction methods to obtain reliable and efficient error estimators. As a simple example, we recover the classical residual error estimators for the second order elliptic equations.
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Coefficient-Robust A Posteriori Error Estimation for H(curl)-elliptic Problems
本文扩展了预处理后验误差估计框架，为H(curl)-椭圆型两相界面问题推导了新的误差估计器，该估计器提供双侧误差界限且对系数变化鲁棒，并与现有方法进行了数值比较。
Deflation-based preconditioning for immersed finite element methods and immersogeometric analysis
本文提出一种基于消元的预处理技术，解决浸没有限元方法中因几何修剪导致的系统矩阵病态问题，通过分析现有预处理策略的局限性，提出更稳健的预处理方法。
Nodal auxiliary a posteriori error estimates
本文在无限维希尔伯特空间中，基于有限元外微积分和节点辅助空间预条件框架，建立后验误差估计与子空间校正预条件子的关键关系，推导真实误差上下界的估计器并应用于多类问题，还提出新正则分解和参数无关估计器。
Some p-robust a posteriori error estimates based on auxiliary spaces
本文基于$H^1$辅助空间分解，为$H(\ m curl)$等三类问题构建p鲁棒平衡后验误差估计，推导了相关方法的误差上界，数值实验验证了估计子的有效性与p鲁棒性。
We modify the well-known interior penalty finite element discretization method so that it allows for element-by-element assembly. This is possible due to the introduction of additional unknowns associated with the interfaces between neighboring elements. The resulting bilinear form, and a Schur complement (reduced) version of it, are utilized in a number of auxiliary space preconditioners for the original conforming finite element discretization problem. These preconditioners are analyzed on the fine scale and their performance is illustrated on model second order scalar elliptic problems disc…
Analysis and preconditioning of parameter-robust finite element methods for Biot's consolidation model
本文研究Biot模型的三场形式，分析参数鲁棒稳定性，提出参数鲁棒误差估计和预处理方法，通过有限元离散化和数值实验验证理论结果。
We describe a fully discrete mixed finite element method for the linearized rotating shallow water model, possibly with damping. While Crank-Nicolson time-stepping conserves energy in the absence of drag or forcing terms and is not subject to a CFL-like stability condition, it requires the inversion of a linear system at each step. We develop weighted-norm preconditioners for this algebraic system that are nearly robust with respect to the physical and discretization parameters in the system. Numerical experiments using Firedrake support the theoretical results.
This paper presents a study of large linear systems resulting from the regular $B$-splines finite element discretization of the $\bm{curl}-\bm{curl}$ and $\bm{grad}-div$ elliptic problems on unit square/cube domains. We consider systems subject to both homogeneous essential and natural boundary conditions. Our objective is to develop a preconditioning strategy that is optimal and robust, based on the Auxiliary Space Preconditioning method proposed by Hiptmair et al. \cite{hiptmair2007nodal}. Our approach is demonstrated to be robust with respect to mesh size, and we also show how it can be com…
Strain smoothing methods such as the smoothed finite element methods (S-FEMs) and the strain-smoothed element~(SSE) method have successfully improved the performance of finite elements, and there have been numerous applications of them in finite element analysis. For the sake of efficient applications to large-scale problems, it is important to develop a mathematically and numerically well-elaborated iterative solver for the strain smoothing methods. In this paper, inspired by the spectral properties of the strain smoothing methods, we propose efficient ways of preconditioning for the methods.…
Optimal preconditioning for the symmetric and non-symmetric coupling of adaptive finite elements and boundary elements
本文研究了适应性耦合FEM与BEM的多级对角加权加法 Schwarz 预处理方法，证明预处理后的系统条件数有界，适用于非对称Johnson-Nédélec单式耦合及其他耦合形式。
Optimal additive Schwarz preconditioning for adaptive 2D IGA boundary element methods
本文研究了适应性二维IGA边界元方法中局部多级对角预条件子的定义与分析，证明了预条件系统条件数与网格尺寸和细化级别无关，从而实现了计算复杂度的最优性。
Numerical simulation of incompressible viscous flow, in particular in three space dimensions, continues to remain a challenging task. Space-time finite element methods feature the natural construction of higher order discretization schemes. They offer the potential to achieve accurate results on computationally feasible grids. Linearizing the resulting algebraic problems by Newton's method yields linear systems with block matrices built of $(k+1)\times (k+1)$ saddle point systems, where $k$ denotes the polynomial order of the variational time discretization. We demonstrate numerically the effi…
Broken-space Additive Schwarz Mass Inverse Approximations and (Block) Preconditioning
研究有限元质量矩阵求解及近似逆问题，提出破碎空间加法 Schwarz（BRAS）质量逆近似方法，证明其对预处理质量矩阵的谱界，通过数值实验表明该方法能降低谱条件数等，在多种有限元及系统上有加速效果。
This paper presents and studies an approach for constructing auxiliary space preconditioners for finite element problems using a constrained nonconforming reformulation, that is based on a proposed modified version of the mortar method. The well-known mortar finite element discretization method is modified to admit a local structure, providing an element-by-element or subdomain-by-subdomain assembly property. This is achieved via the introduction of additional trace finite element spaces and degrees of freedom (unknowns) associated with the interfaces between adjacent elements or subdomains. T…
We devise 3-field and 4-field finite element approximations of a system describing the steady state of an incompressible heat-conducting fluid with implicit non-Newtonian rheology. We prove that the sequence of numerical approximations converges to a weak solution of the problem. We develop a block preconditioner based on augmented Lagrangian stabilisation for a discretisation based on the Scott-Vogelius finite element pair for the velocity and pressure. The preconditioner involves a specialised multigrid algorithm that makes use of a space-decomposition that captures the kernel of the diverge…
Multigrid Preconditioning for FD-DLM Method in Elliptic Interface Problems
本文研究FD-DLM方法求解椭圆界面问题中线性系统的多重网格预处理器性能，实验显示其作为预处理器的有效性。
This work proposes an original preconditioner that couples the Constrained Pressure Residual (CPR) method with block preconditioning for the efficient solution of the linearized systems of equations arising from fully implicit multiphase flow models. This preconditioner, denoted as Block CPR (BCPR), is specifically designed for Lagrange multipliers-based flow models, such as those generated by Mixed Hybrid Finite Element (MHFE) approximations. An original MHFE-based formulation of the two-phase flow model is taken as a reference for the development of the BCPR preconditioner, in which the set …
In this paper we introduce and analyze the residual-based a posteriori error estimation of the partially penalized immersed finite element method for solving elliptic interface problems. The immersed finite element method can be naturally utilized on interface-unfitted meshes. Our a posteriori error estimate is proved to be both reliable and efficient with reliability constant independent of the location of the interface. Numerical results indicate that the efficiency constant is independent of the interface location and that the error estimation is robust with respect to the coefficient contr…
A class of linear parabolic equations are considered. We give a posteriori error estimates in the maximum norm for a method that comprises extrapolation applied to the backward Euler method in time and finite element discretisations in space. We use the idea of elliptic reconstructions and certain bounds for the Green's function of the parabolic operator.
Recovery type a posteriori error estimation of an adaptive finite element method for Cahn--Hilliard equation
该研究针对Cahn--Hilliard方程的Crank-Nicolson有限元方法，结合椭圆与时间重构技术推导了新型恢复型后验误差估计，构建时空自适应算法，并通过数值实验验证了理论结果的有效性。
In this paper we develop a new a posteriori error analysis for the Monge-Ampère equation approximated by conforming finite element method on isotropic meshes in 2D. The approach utilizes a slight variant of the mixed discretization proposed by Gerard Awanou and Hengguang Li in International Journal of Numerical Analysis and Modeling, 11(4):745-761, 2014. The a posteriori error estimate is based on a suitable evaluation on the residual of the finite element solution. It is proven that the a posteriori error estimate provided in this paper is both reliable and efficient.
A posteriori error analysis for a continuous space-time finite element method for a hyperbolic integro-differential equation
针对具有混合边界条件的双曲型积分微分方程，提出了一种一次连续时空有限元方法，并基于时空单元给出了后验误差表示，可用于对偶加权残差法的自适应策略，同时证明了基于加权全局投影和局部投影的后验误差估计。
Preconditioning of a hybridized discontinuous Galerkin finite element method for the Stokes equations
本文研究了斯托克斯方程的混合间断有限元方法的预条件化问题，通过静态消元技术减少全局问题的规模，并证明了压力Schur补与两个简单矩阵的谱等价性，从而提出最优预条件化方法。
Residual-Based A Posteriori Error Estimates for Symmetric Conforming Mixed Finite Elements for Linear Elasticity Problems
本文提出用于处理Dirichlet和混合边界条件的线弹性问题对称混合有限元的后验误差估计方法，证明了其稳定性和效率，并通过数值实验验证理论结果。
In this paper, a residual-type a posteriori error estimator is proposed and analyzed for a modified weak Galerkin finite element method solving linear elasticity problems. The estimator is proven to be both reliable and efficient because it provides upper and lower bounds on the actual error in a discrete energy norm. Numerical experiments are given to illustrate the effectiveness of the this error estimator.
A posteriori error estimators suitable for moving finite element methods under anisotropic meshes
针对一般二阶椭圆问题，提出一种适用于各向异性网格下移动有限元方法的新型后验误差估计子，通过恢复Hessian矩阵简化计算，数值实验展示出优异的效率指标。
A posteriori estimates for conforming Kirchhoff plate elements
本文研究了Kirchhoff板弯曲问题的后验估计，考虑了混合边界条件和多种载荷情况，并通过大量数值计算验证了估计器的有效性。
A posteriori error estimates for a Virtual Elements Method for the Steklov eigenvalue problem
本文研究了虚拟元方法在Steklov特征值问题中的后验误差分析，提出了一种残差型误差估计器，并通过数值实验验证了其有效性。
We derive an anisotropic a posteriori error estimate for the adaptive conforming Virtual Element approximation of a paradigmatic two-dimensional elliptic problem. In particular, we introduce a quasi-interpolant operator and exploit its approximation results to prove the reliability of the error indicator. We design and implement the corresponding adaptive polygonal anisotropic algorithm. Several numerical tests assess the superiority of the proposed algorithm in comparison with standard polygonal isotropic mesh refinement schemes.
Robust topology optimization using a posteriori error estimator for the finite element method
本文提出利用后验误差估计器改进密度法，以防止棋盘效应并提升解的正则性和结果鲁棒性。
An explicit and computable error estimator for the $hp$ version of the virtual element method (VEM), together with lower and upper bounds with respect to the exact energy error, is presented. Such error estimator is employed to provide $hp$ adaptive mesh refinements for very general polygonal meshes. In addition, a novel VEM $hp$ Clément quasi-interpolant, instrumental for the a posteriori error analysis, is introduced. The performances of the adaptive method are validated by a number of numerical experiments.
Finite Element Methods for Interface Problems: Robust Residual-Based A Posteriori Error Estimates
本文研究了椭圆界面问题中不同有限元近似方法的残差型后验误差估计，建立了与扩散系数跃变无关的全局可靠性界，并无需假设扩散系数分布。
The high-order finite element Duffy de Rham complex and low-order-refined preconditioning
本文构建了适用于低阶精细化预条件化的高阶有限元空间，利用Duffy变换将多项式空间拉回至单位正方形，同时包含有理函数，可退化为标准拉格朗日、Nédélec和间断有限元。通过Jacobi加权L²范数证明了刚度矩阵的谱等价性，并通过数值实验验证了预条件化效果。
In two dimensions, we propose and analyze an a posteriori error estimator for the acoustic spectral problem based on the virtual element method in $\H(÷;Ω)$. Introducing an auxiliary unknown, we use the fact that the primal formulation of the acoustic problem is equivalent to a mixed formulation, in order to prove a superconvergence result, necessary to despise high order terms. Under the virtual element approach, we prove that our local indicator is reliable and globally efficient in the $Ł^2$-norm. We provide numerical results to assess the performance of the proposed error estimator.
Algebraic Hybridization and Static Condensation with Application to Scalable H(div) Preconditioning
本文提出一种统一的代数方法用于静态凝结和混合，通过数值实验展示了两种技术相对于H(div)空间的可扩展求解器的性能，证明混合技术在高阶元素中表现更优。
Least-Squares versus Partial Least-Squares Finite Element Methods: Robust A Priori and A Posteriori Error Estimates of Augmented Mixed Finite Element Methods
本文针对广义达西问题，提出两种稳健的增广混合有限元方法（偏最小二乘方法），与标准LSFEM共享后验误差估计器，并首次证明基于L2的LSFEM的稳健性，数值实验验证了结果。
Localized pointwise a posteriori error estimates for nonconforming finite element methods
针对泊松和双调和方程的非协调有限元离散，本文设计新型权重函数，推导Crouzeix--Raviart方法及Morley离散的局部后验误差估计，实现对相关误差的精确控制。
Preconditioning immersed isogeometric finite element methods with application to flow problems
本文提出了一种基于连接性的预处理技术，用于解决非对称正定问题和混合问题中的浸入式有限元方法的条件问题，通过数值实验验证了该预处理方法的有效性。
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。
A posteriori error analysis for finite element solution of elliptic differential equations using equidistributing meshes
针对线性椭圆微分方程，提出基于残差后验误差估计的等分布网格自适应有限元方法，并严格分析线性有限元近似的收敛性。
In this paper, we propose a new finite element solution approach to the multi-compartmental Darcy equations describing flow and interactions in a porous medium with multiple fluid compartments. We introduce a new numerical formulation and a block-diagonal preconditioner. The robustness with respect to variations in material parameters is demonstrated by theoretical considerations and numerical examples.
This is the third part in a series on a mass conserving, high order, mixed finite element method for Stokes flow. In this part, we study a block-diagonal preconditioner for the indefinite Schur complement system arising from the discretization of the Stokes equations using these elements. The underlying finite element method is uniformly stable in both the mesh size h and polynomial order p, and we prove bounds on the eigenvalues of the preconditioned system which are independent of h and grow modestly in p. The analysis relates the Schur complement system to an appropriate variational setting…
Explicit implementation strategy of high order edge finite elements and Schwarz preconditioning for the time-harmonic Maxwell's equations
本文研究了高阶有限元在时谐Maxwell方程中的应用，提出显式实现策略及Schwarz预条件化方法，解决波导配置中的计算问题。
Generalised gradients for virtual elements and applications to a posteriori error analysis
本文将标准节点虚拟元方法改写为广义梯度方法，通过局部重构间断通量与势构造可靠高效的误差估计子，并证明其上下界常数独立于稳定化及网格假设下的精度次数。
Two asymptotically exact a posteriori error estimates are proposed for eigenvalues by the nonconforming Crouzeix--Raviart and enriched Crouzeix-- Raviart elements. The main challenge in the design of such error estimators comes from the nonconformity of the finite element spaces used. Such nonconformity causes two difficulties, the first one is the construction of high accuracy gradient recovery algorithms, the second one is a computable high accuracy approximation of a consistency error term. The first difficulty was solved for both nonconforming elements in a previous paper. Two methods are …
A residual a posteriori error estimate for the Stabilization-free Virtual Element Method
本文提出了一种无稳定化虚拟元方法的后验误差估计，通过证明误差度量与残差估计器的等价性，验证了其在不同网格和扩散项跃变下的鲁棒性。
A Posteriori Error Estimates with Boundary Correction for a Cut Finite Element Method
本文为CutFEM方法提出带边界修正的后验误差估计，考虑非多边形边界及几何与数据逼近，理论证明可靠性与效率，且常数对网格切割方式鲁棒。
We develop structure-preserving finite element methods for the incompressible, resistive Hall magnetohydrodynamics (MHD) equations. These equations incorporate the Hall current term in Ohm's law and provide a more appropriate description of fully ionized plasmas than the standard MHD equations on length scales close to or smaller than the ion skin depth. We introduce a stationary discrete variational formulation of Hall MHD that enforces the magnetic Gauss's law exactly (up to solver tolerances) and prove the well-posedness and convergence of a Picard linearization. For the transient problem, …
An example of explicit implementation strategy and preconditioning for the high order edge finite elements applied to the time-harmonic Maxwell's equations
本文研究了高阶有限元近似电场与合适预处理器结合，用于求解波导配置中的时谐Maxwell方程。重点在于高阶curl-一致有限元的实现，特别是三维情况下的复杂性，并通过FreeFem++软件展示了实现策略。同时探讨了时谐Maxwell方程的预处理方法，尤其针对高阶离散化。
Smoother-type a posteriori error estimates for finite element methods
本文提出基于平滑器的有限元后验误差估计方法，通过简单平滑器处理残差实现高精度和鲁棒性，优于传统残差型估计器。
Residual-Based a posteriori error estimation for hp-adaptive finite element methods for the Stokes equations
本文提出了一种基于残差的后验误差估计方法，用于斯托克斯方程的hp自适应有限元方法，通过扩展经典h版本AFEM的后验误差估计思想，建立了误差估计器的可靠性和有效性，并通过数值实验验证了其性能。
Supremum norm A Posteriori Error control of Quadratic Finite Element Method for the Signorini problem
本文针对Signorini问题提出二次有限元方法的逐点后验误差估计器，通过构造接触力密度离散对应物及引入障碍函数实现极大范误差控制，数值实验验证了其优异性能。
Low-Order Refined Preconditioning for Spectral/hp Element Method for Complex, 3D Geometries
本文扩展低阶细化预处理至单纯形和混合单元，建立广义范德蒙德变换，在复杂三维几何中显著减少CG迭代次数并加速生产级模拟。
Residual-based a posteriori error estimation for multipoint flux mixed finite element methods
本文提出了一种新型残差型后验误差分析技术，用于二维或三维多点通量混合有限元方法中的流体流动问题。所提出的后验误差估计器包含离散化和求积指示量，能够可靠且高效地估计速度和压力的L²范数误差。
Assessing Finite Element Choice in Structural Topology Optimization and A Posteriori Error Estimation
本文研究了有限元选择对结构拓扑优化的影响，采用SIMP方法，比较了线性（P1）和二次（P2）三角元与传统双线性四边形（Q1）元在基准问题中的表现，揭示了元素类型对优化目标值和解精度的影响。
Functional error estimates are well-established tools for a posteriori error estimation and related adaptive mesh-refinement for the finite element method (FEM). The present work proposes a first functional error estimate for the boundary element method (BEM). One key feature is that the derived error estimates are independent of the BEM discretization and provide guaranteed lower and upper bounds for the unknown error. In particular, our analysis covers Galerkin BEM and the collocation method, what makes the approach of particular interest for scientific computations and engineering applicati…
Convergence analysis of iterative solution with inexact block preconditioning for weak Galerkin finite element approximation of Stokes flow
本文分析了采用weak Galerkin有限元离散并使用非精确块Schur补预处理的Stokes流迭代解收敛性，证明MINRES/GMRES的收敛因子几乎与粘度和网格大小无关。
Residual-based a Posteriori Error Estimate for Interface Problems: Nonconforming Linear Elements
本文研究了非匹配线性有限元近似界面问题的改进残差后验误差估计器，通过新方法证明其可靠性，无需使用霍尔姆兹分解，且不假设扩散系数为近单调。
Residual-based a posteriori error estimates for a conforming finite element discretization of the Navier-Stokes/Darcy coupled problem
本文提出一种新的后验残差型误差估计器，用于Navier-Stokes与多孔介质流耦合问题的符合混合有限元方法，通过验证其可靠性和有效性，扩展至其他有限元子空间以获得稳定Galerkin方案。
Parameter-free inexact block Schur complement preconditioning for linear poroelasticity under a hybrid Bernardi-Raugel and weak Galerkin finite element discretization
本文提出一种无需参数的近似块Schur补预处理方法，用于解决线性多孔弹性问题，通过混合Bernardi-Raugel和弱格架有限元离散化，实现对锁紧现象的正则化处理，提高求解效率和鲁棒性。
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