Pablo Montiel López
In this work, we characterize the existence of solution for a certain variational inequality by means of a classical minimax theorem. In addition, we propose a numerical algorithm for the solution of an inverse problem associated with a variational inequality. To this end we state a collage-type result in this variational framework.
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In many applications of practical interest, solutions of partial differential equation models arise as critical points of an underlying (energy) functional. If such solutions are saddle points, rather than being maxima or minima, then the theoretical framework is non-standard, and the development of suitable numerical approximation procedures turns out to be highly challenging. In this paper, our aim is to present an iterative discretization methodology for the numerical solution of nonlinear variational problems with multiple (saddle point) solutions. In contrast to traditional numerical appr…
Variational-hemivariational inequalities: A brief survey on mathematical theory and numerical analysis
本文综述了变分-半变分不等式的数学理论和数值分析，探讨了其在物理和工程中的应用及求解方法。
Well-posedness and Numerical Analysis of Mixed Variational-hemivariational Inequalities
本文研究了混合变分-半变分不等式的适定性及数值解法，并推导了最优阶误差估计，通过数值实验验证了收敛阶的理论预测。
A new Discrete Analysis Of Fourth Order Elliptic Variational Inequalities
本文将梯度离散化方法（GDM）用于四阶椭圆变分不等式，建立新的误差估计与完整收敛性分析，并证明在经典数据假设下收敛是无条件的，适用于所有 GDM 框架内的数值格式。
Semicoercive Variational Inequalities - From Existence to Numerical Solution of Nonmonotone Contact Problems
本文提出了解半 coercive 半变分不等式的新型数值解法，并通过非单调摩擦的单向接触问题作为模型示例，展示了基准测试的数值结果。
Additive Schwarz methods for fourth-order variational inequalities
本文研究四阶变分不等式的加性Schwarz方法，基于统一有限元框架开发了一层和两层方法，并利用新型非线性保正粗插值算子首次证明了两层方法的可扩展性，数值实验验证了理论结果。
The proximal Galerkin method for non-symmetric variational inequalities
本文提出了一种非对称变分不等式的近邻伽辽金方法，通过建立符合性和混合混合一阶系统变体，实现了最优误差估计，并应用于美式期权定价等实际问题。
The gradient discretisation method (GDM) is a generic framework designed recently, as a discretise in spatial space, to partial differential equations. This paper aims to use the GDM to establish a first general error estimate for numerical approximations of parabolic obstacle problems. This gives the convergence rates of several well--known conforming and non conforming numerical methods. Numerical experiments based on the hybrid finite volume method are provided to verify the theoretical results.
This paper is concerned with modeling the polarization process in ferroelectric media. We develop a thermodynamically consistent model, based on phenomenological descriptions of free energy as well as switching and saturation conditions in form of inequalities. Thermodynamically consistent models naturally lead to variational formulations. We propose to use the concept of variational inequalities. We aim at combining the different phenomenological conditions into one variational inequality. In our formulation we use one Lagrange multiplier for each condition (the onset of domain switching and …
This work aims to provide a comprehensive and unified numerical analysis for non linear system of parabolic variational inequalities (PVIs) subject to Dirichlet boundary condition. This analysis enables us to establish an existence of the exact solution to the considered model and to prove the convergence for the approximate solution and its approximate gradient. Our results are applicable for several conforming and non conforming numerical schemes.
In this article we consider a method for proving a class of analytical inequalities via minimax rational approximations. All numerical calculations in this paper are given by Maple computer program.
We propose an approximation scheme for a class of semilinear variational inequalities whose Hamiltonian is convex and coercive. The proposed scheme is a natural extension of a previous splitting scheme proposed by Liang, Zariphopoulou and the author for semilinear parabolic PDEs. We establish the convergence of the scheme and determine the convergence rate by obtaining its error bounds. The bounds are obtained by Krylov's shaking coefficients technique and Barles-Jakobsen's optimal switching approximation, in which a key step is to introduce a variant switching system.
Exterior Nonlocal Variational Inequalities
本文提出外部非局部变分不等式，利用分数阶算子将障碍物置于观测域之外，克服经典方法只能将障碍物置于域内或边界上的局限，并给出连续问题分析与扰动逼近方法。
Variational Inequalities and Improved Convergence Rates for Tikhonov Regularisation on Banach Spaces
针对Banach空间上的线性算子方程，利用对偶泛函的变分不等式，推导出Tikhonov型凸正则化在Bregman距离下的高阶收敛速率，突破了传统低阶限制。
Since its introduction over 50 years ago, the concept of Mosco convergence has permeated through diverse areas of mathematics and applied sciences. These include applied analysis, the theory of partial differential equations, numerical analysis, and infinite dimensional constrained optimization, among others. In this paper we explore some of the consequences of Mosco convergence on applied problems that involve moving sets, with some historical accounts, and modern trends and features. In particular, we focus on connections with density of convex intersections, finite element approximations, q…
Variational inequalities of multilayer elastic systems with interlayer friction: existence and uniqueness of solution and convergence of numerical solution
本文构建了具层间摩擦的多层弹性系统模型，推导其变分不等式，证明了解的存在唯一性及有限元数值解的收敛性与收敛阶，为位移-应变问题提供理论支撑。
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