A New Approach to Source Conditions in Regularization with General Residual Term

This article addresses Tikhonov-like regularization methods with convex penalty functionals for solving nonlinear ill-posed operator equations formulated in Banach or, more general, topological spaces. We present an approach for proving convergence rates that combines advantages of approximate source conditions and variational inequalities. Precisely, our technique provides both a wide range of convergence rates and the capability to handle general and not necessarily convex residual terms as well as nonsmooth operators. Initially formulated for topological spaces, the approach is extensively discussed for Banach and Hilbert space situations, showing that it generalizes some well-known convergence rates results.     

A Posteriori Error Estimates of Mixed Methods for Parabolic Optimal Control Problems

In this article, we shall give a brief review on the fully discrete mixed finite element method for general optimal control problems governed by parabolic equations. The state and the co-state are approximated by the lowest order Raviart–Thomas mixed finite element spaces and the control is approximated by piecewise constant elements. Furthermore, we derive a posteriori error estimates for the finite element approximation solutions of optimal control problems. Some numerical examples are given to demonstrate our theoretical results.     

Superconvergence of H 1-Galerkin Mixed Finite Element Methods for Second-Order Elliptic Equations

We derive superconvergence result for H ¹-Galerkin mixed finite element method for second-order elliptic equations over rectangular partitions. Compared to standard mixed finite element procedure, the method is not subject to the Ladyzhenskaya–Bab?ska–Brezzi (LBB) condition and the approximating finite element spaces are allowed to be of different polynomial degrees. Superconvergence estimate of order 𝒪(h ^k+3), where k ≥ 1 is the order of the approximating polynomials employed in the Raviart–Thomas elements, is established for the flux via a postprocessing technique.     

椭圆边界上的自然积分算子及各向异性外问题的耦合算法

1.引 言为求解微分方程的外边值问题常需要引进人工边界（见[1-4]）,对人工边界外部区域作自然边界归化得到的自然积分方程即Dirichlet-Neumann映射,正是人工边界上的准确的边界条件（见[2-6]）,这是一类非局部边界条件.自然积分算子即Dirichlet-Neumann算子,     

The semiclassical modified nonlinear Schrödinger equation I: Modulation theory and spectral analysis

We study an integrable modification of the focusing nonlinear Schrödinger equation from the point of view of semiclassical asymptotics. In particular, (i) we establish several important consequences of the mixed-type limiting quasilinear system including the existence of maps that embed the limiting forms of both the focusing and defocusing nonlinear Schrödinger equations into the framework of a single limiting system for the modified equation, (ii) we obtain bounds for the location of the discrete spectrum for the associated spectral problem that are particularly suited to the semiclassical limit and that generalize known results for the spectrum of the nonselfadjoint Zakharov-Shabat spectral problem, and (iii) we present a multiparameter family of initial data for which we solve the associated spectral problem in terms of special functions for all values of the semiclassical scaling parameter. We view our results as part of a broader project to analyze the semiclassical limit of the modified nonlinear Schrödinger equation via the noncommutative steepest descent procedure of Deift and Zhou, and we also present a selfcontained development of a Riemann-Hilbert problem of inverse scattering that differs from those given in the literature and that is well adapted to semiclassical asymptotics.     

瞬态热传导问题的复变量重构核粒子法

在重构核粒子法的基础上,引入复变量,讨论了复变量重构核粒子法.复变量重构核粒子法的优点是在构造形函数时采用一维基函数建立二维问题的修正函数.然后,将复变量重构核粒子法应用于瞬态热传导问题的求解,结合瞬态热传导问题的Galerkin积分弱形式,采用罚函数法引入本质边界条件,建立了瞬态热传导问题的复变量重构核粒子法,推导了相应的计算公式.与传统的重构核粒子法相比,复变量重构核粒子法具有计算量小、精度高的优点.最后通过数值算例证明了该方法的有效性. 关键词： 重构核粒子法 复变量重构核粒子法 修正函数 瞬态热传导问题     

弹性矩形板方程在非线性边界条件下整体解的存在唯一性

考虑了在非线性边界条件下的弹性矩形板方程.利用Galerkin方法,首先证明了该方程在非线性边界(a)及初值w⁰∈W,w¹∈W的条件下初边值问题存在唯一整体弱解w(t).其次证明了该方程在非线性边界(b)及初值w⁰∈W₁,w¹∈W₁的条件 关键词： 弹性矩形板方程 非线性边界条件 初边值问题 整体解