Susanne C. Brenner
Finite element methods for a model elliptic distributed optimal control problem with pointwise state constraints are considered from the perspective of fourth order boundary value problems.
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Discontinuous Galerkin Methods for an Elliptic Optimal Control Problem with a General State Equation and Pointwise State Constraints
针对一般多边形域上带逐点状态约束的椭圆最优控制问题，提出并分析间断伽辽金方法，建立误差估计并通过数值实验验证。
We consider a one dimensional elliptic distributed optimal control problem with pointwise constraints on the derivative of the state. By exploiting the variational inequality satisfied by the derivative of the optimal state, we obtain higher regularity for the optimal state under appropriate assumptions on the data. We also solve the optimal control problem as a fourth order variational inequality by a $C^1$ finite element method, and present the error analysis together with numerical results.
New Error Estimates for An Elliptic Distributed Optimal Control Problem with Pointwise Control Constraints
针对带逐点控制约束的线性二次椭圆型分布最优控制问题，推导了可同时适用于标准有限元方法和多尺度有限元方法的新误差估计。
A New Error Analysis for a Point Tracking Elliptic Distributed Optimal Control Problem with Pointwise Control Constraints
本文针对带点跟踪和点态控制约束的线性二次椭圆型分布最优控制问题，在统一框架下对标准有限元离散与变分离散开展误差分析，其自包含性依托有限元分析标准技术与椭圆型偏微分方程经典结果。
A New Error Analysis for Finite Element Methods for Elliptic Neumann Boundary Control Problems with Pointwise Control Constraints
该研究针对带Neumann边界控制与逐点控制约束的线性二次椭圆最优控制问题，提出一种适用于光滑系数下标准有限元、粗糙系数下多尺度有限元的新误差分析方法。
Pointwise A Posteriori Error Estimators for Elliptic Eigenvalue Problems
本文针对椭圆特征值问题提出并分析了逐点残差型后验误差估计子，证明其在 $L^{\infty}$ 范数下的可靠性与有效性，并将方法推广到非协调有限元逼近，数值实验验证了理论结果。
We present a posteriori error analysis in the supremum norm for the symmetric interior penalty discontinuous Galerkin method for the elliptic obstacle problem. We construct discrete barrier functions based on appropriate corrections of the conforming part of the solution that is obtained via a constrained averaging operator. The corrector function accounts properly for the non-conformity of the approximation and it is estimated by direct use of the Green's function of the unconstrained elliptic problem. The use of the continuous maximum principle guarantees the validity of the analysis without…
We consider a pointwise tracking optimal control problem for a semilinear elliptic partial differential equation. We derive the existence of optimal solutions and analyze first and, necessary and sufficient, second order optimality conditions. We devise two strategies of discretization to approximate a solution of the optimal control problem: a semidiscrete scheme where the control variable is not discretized -- the so-called variational discretization approach -- and a fully discrete scheme where the control variable is discretized with piecewise constant functions. For both solution techniqu…
A new finite element method for elliptic optimal control problems with pointwise state constraints in energy spaces
针对带逐点状态约束的椭圆型分布与边界控制问题，提出利用能量空间范数等价表示将其转化为仅含状态的二阶变分不等式，并用标准 $C^0$ 有限元求解，给出误差估计与数值验证。
We investigate $C^1$ finite element methods for one dimensional elliptic distributed optimal control problems with pointwise constraints on the derivative of the state formulated as fourth order variational inequalities for the state variable. For the problem with Dirichlet boundary conditions, we use an existing $H^{\frac52-ε}$ regularity result for the optimal state to derive $O(h^{\frac12-ε})$ convergence for the approximation of the optimal state in the $H^2$ norm. For the problem with mixed Dirichlet and Neumann boundary conditions, we show that the optimal state belongs to $H^3$ under ap…
Finite element error estimates for a bilinear optimal control problem with pointwise tracking governed by a semilinear elliptic PDE
针对半线性椭圆PDE控制的逐点跟踪双线性最优控制问题，提出半离散与全离散有限元方案，证明收敛性及误差界，二维下误差改进，数值实验验证性能。
The main goal of the paper is to show new stability and localization results for the finite element solution of the Stokes system in $W^{1,\infty}$ and $L^{\infty}$ norms under standard assumptions on the finite element spaces on quasi-uniform meshes in two and three dimensions. Although interior error estimates are well-developed for the elliptic problem, they appear to be new for the Stokes system on unstructured meshes. To obtain these results we extend previously known stability estimates for the Stokes system using regularized Green's functions.
A Multiscale Finite Element Method for an Elliptic Distributed Optimal Control Problem with Rough Coefficients and Control Constraints
针对含粗糙系数和逐点控制约束的椭圆型分布最优控制问题，提出并分析一种多尺度有限元方法，证明其性能可媲美标准有限元在光滑问题上的表现，并用数值实验验证。
Finite Element Error Estimates for Optimal Control Problems with Pointwise Tracking
本文研究了具有状态变量点评估的线性二次椭圆最优控制问题，通过有限元方法改进了已知的先验离散误差估计，推导了不同控制离散化方法的最优误差估计。
Additive Schwarz preconditioners for a state constrained elliptic distributed optimal control problem discretized by a partition of unity method
本文研究了使用分区统一方法离散化的椭圆分布最优控制问题，提出加性Schrödinger预条件子，并通过数值示例验证了条件数估计。
Pointwise A Posteriori Error Estimators for Multiple and Clustered Eigenvalue Computations
针对椭圆算子多特征值和聚类特征值的特征函数有限元逼近，提出可靠高效且常数与特征值间隙等无关的点态后验误差估计器，揭示线性单元边残差主导性并通过数值实验验证。
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。
Discrete maximum principles for nonlinear elliptic finite element problems on Riemannian manifolds with boundary
本文研究了在带边黎曼流形上非线性表面有限元问题的离散最大原理，通过类比Pucci等人经典点wise最大原理，建立了相应的离散最大原理，并通过实例展示了结果的应用范围。
A Nitsche-eXtended finite element method for distributed optimal control problems of elliptic interface equations
本文提出了一种适用于椭圆界面方程分布式最优控制问题的界面不匹配数值方法，通过变分离散化概念对最优控制问题进行离散化，并应用Nitsche扩展有限元方法对相应状态和伴随方程进行离散化，通过在界面周围引入分段切割基函数来增强标准线性单元空间，推导出状态、伴随状态和控制在网格依赖范数和L²范数下的最优误差估计，并通过数值结果验证理论结果。
An EDG Method for Distributed Optimal Control of Elliptic PDEs
本文提出EDG方法用于求解椭圆PDE的分布式最优控制问题，推导了状态、共轭状态、最优控制的最优先验误差估计，并通过数值实验验证理论结果。
This article is concerned with the nonconforming finite element method for distributed elliptic optimal control problems with pointwise constraints on the control and gradient of the state variable. We reduce the minimization problem into a pure state constraint minimization problem. In this case, the solution of the minimization problem can be characterized as fourth-order elliptic variational inequalities of the first kind. To discretize the control problem we have used the bubble enriched Morley finite element method. To ensure the existence of the solution to discrete problems three bubble…
Numerical analysis of a family of optimal distributed control problems governed by an elliptic variational inequality
本文研究了由参数α>0的椭圆变分不等式所 governed 的最优分布式控制问题家族，通过有限元方法分析其数值特性，并探讨了参数h→0和α→∞时的极限行为及收敛性。
Pointwise best approximation results for Galerkin finite element solutions of parabolic problems
本文研究了抛物问题在L^∞范数下的Galerkin有限元解的最佳逼近性质，采用空间连续拉格朗日有限元和时间间断Galerkin方法，首次在三维空间中获得此类结果，且证明不依赖空间和时间网格尺寸的关系。
Adaptive finite element method for elliptic optimal control problems: convergence and optimality
本文研究了带点约束的椭圆最优控制问题的自适应有限元方法收敛性，通过变分离散化和线性连续有限元逼近，证明了方法对状态和伴随状态的收敛性和近似最优性。
Global and interior pointwise best approximation results for the gradient of Galerkin solutions for parabolic problems
本文研究了抛物问题全离散格雷戈尔解在$ L^\infty(I;W^{1,\infty}(\Om)) $范数下的最佳逼近性质，采用空间连续拉格朗日有限元和时间任意阶间断伽辽金方法，证明方法不同于现有全离散误差估计技术，仅使用椭圆结果和离散最大抛物正则性。
A monotone finite element method for an elliptic distributed optimal control problem with a convection-dominated state equation
本文提出了一种单调有限元方法，用于求解对流主导的椭圆分布最优控制问题，通过保持离散最大原理和最优收敛阶次，确保数值稳定性与鲁棒性。
Numerical analysis of distributed optimal control problems governed by elliptic variational inequalities
本文基于椭圆变分不等式构建最优控制问题，利用有限元方法分析其数值解的存在性与收敛性，证明离散解向连续解收敛。
Mixed Consistent PINNs for Elliptic Obstacle Problems with Stability Analysis
本文提出了一种用于椭圆障碍问题的混合一致物理信息神经网络框架，通过混合损失函数和一致的拉格朗日乘子处理，实现了解和乘子的近最优收敛率，展示了方法的准确性、稳定性和鲁棒性。
Numerical solution of elliptic distributed optimal control problems with boundary value tracking
本文提出了一种基于张量积有限元的椭圆分布最优控制问题求解方法，通过数值实验验证了理论结果的准确性。
We consider standard tracking-type, distributed elliptic optimal control problems with $L^2$ regularization, and their finite element discretization. We are investigating the $L^2$ error between the finite element approximation $u_{\varrho h}$ of the state $u_\varrho$ and the desired state (target) $\bar{u}$ in terms of the regularization parameter $\varrho$ and the mesh size $h$ that leads to the optimal choice $\varrho = h^4$. It turns out that, for this choice of the regularization parameter, we can devise simple Jacobi-like preconditioned MINRES or Bramble-Pasciak CG methods that allow us …
Mass-lumping discretization and solvers for distributed elliptic optimal control problems
针对带$L_2$正则化的分布式椭圆最优控制问题，研究质量集中在有限元离散中的效果，证明其不影响$L_2$误差且可构建快速Schur补系统，结合嵌套迭代的预处理共轭梯度法得到渐近最优求解器。
A reconstructed discontinuous approximation for distributed elliptic control problems
本文提出了一种用于分布式椭圆最优控制问题的内部惩罚不连续伽辽金方法，实现了高阶近似空间和高效求解的离散系统，并提供了误差估计和数值验证。
Large linear systems of saddle-point type have arisen in a wide variety of applications throughout computational science and engineering. The discretizations of distributed control problems have a saddle-point structure. The numerical solution of saddle-point problems has attracted considerable interest in recent years. In this work, we propose a novel Braess-Sarazin multigrid relaxation scheme for finite element discretizations of the distributed control problems, where we use the stiffness matrix obtained from the five-point finite difference method for the Laplacian to approximate the inver…
Regularization and finite element error estimates for elliptic distributed optimal control problems with energy regularization and state or control constraints
本文研究了带状态或控制约束及能量正则化的椭圆分布最优控制问题，通过关联正则化参数与有限元网格尺寸确保最优收敛阶，并建立第一类变分不等式求解，数值结果验证了方法的准确性与适用性。
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