A One Dimensional Elliptic Distributed Optimal Control Problem with Pointwise Derivative Constraints

Pointwise A Posteriori Error Analysis of a Discontinuous Galerkin Method for the Elliptic Obstacle Problem

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…

Error estimates for a pointwise tracking optimal control problem of a semilinear elliptic equation

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…

能量空间中带逐点状态约束的椭圆型最优控制问题的一种新有限元方法

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。

Finite Element Methods for One Dimensional Elliptic Distributed Optimal Control Problems with Pointwise Constraints on the Derivative of the State

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…

分数物质导数：逐点表示与有限体积数值格式

Fractional material derivative: pointwise representation and a finite volume numerical scheme

本文给出分数物质导数的时空逐点局部表示，分析相关方程解的存在性，并提出稳定、收敛且保持概率守恒的有限体积格式，计算效率优于 Monte Carlo 方法。

Pointwise-in-time a posteriori error control for time-fractional parabolic equations

For time-fractional parabolic equations with a Caputo time derivative of order $α\in(0,1)$, we give pointwise-in-time a posteriori error bounds in the spatial $L_2$ and $L_\infty$ norms. Hence, an adaptive mesh construction algorithm is applied for the L1 method, which yields optimal convergence rates $2-α$ in the presence of solution singularities.

各向异性网格上具混合导数椭圆问题单调离散化的一个反例

Monotone discretization of elliptic problems with mixed derivatives on anisotropic meshes: a counterexample

本文提出一个反例，探讨各向异性网格上具混合导数椭圆问题的单调离散化。

半线性椭圆偏微分方程控制的具有逐点跟踪的双线性最优控制问题的有限元误差估计

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。

Global and local pointwise error estimates for finite element approximations to the Stokes problem on convex polyhedra

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.

Pointwise-in-time a posteriori error control for higher-order discretizations of time-fractional parabolic equations

Time-fractional parabolic equations with a Caputo time derivative are considered. For such equations, we explore and further develop the new methodology of the a-posteriori error estimation and adaptive time stepping proposed in [7]. We improve the earlier time stepping algorithm based on this theory, and specifically address its stable and efficient implementation in the context of high-order methods. The considered methods include an L1-2 method and continuous collocation methods of arbitrary order, for which adaptive temporal meshes are shown to yield optimal convergence rates in the presen…

一种求解具有粗糙系数和控制约束的椭圆型分布最优控制问题的多尺度有限元方法

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。

有限元误差估计：具有点跟踪的最优控制问题

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。