A Sequential Minimization Technique for Elliptic Quasi-variational Inequalities with Gradient Constraints
A Sequential Minimization Technique for Elliptic Quasi-variational Inequalities with Gradient Constraints
复制标题
具有梯度约束的椭圆拟变分不等式的序贯最小化技术
DOI：
10.1137/110837048
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
C. N. Rautenberg
中科院分区：
文献类型：
--
作者：
M. Hintermüller;C. N. Rautenberg
研究了函数空间中一类带梯度约束的非线性椭圆型拟变分不等式问题。例如，在超导体磁化的数学描述中，在弹塑性问题中，或在静电学以及博弈论中，都会出现这类问题。本文利用变分不等式型问题的迭代解，利用序列极小化技术求出了QVI的迭代解。一个单调算子理论的方法，不诉诸Mosco收敛，因为经常做的存在性分析QVI。对于QVI的数值解的惩罚方法结合半光滑牛顿迭代提出。本文最后的报告涉及$p$-拉普拉斯算子和各种类型的非线性约束的数值试验。
A class of nonlinear elliptic quasi-variational inequality (QVI) problems with gradient constraints in function space is considered. Problems of this type arise, for instance, in the mathematical description of the magnetization of superconductors, in problems in elastoplasticity, or in electrostatics as well as in game theory. The paper addresses the iterative solution of the QVIs by a sequential minimization technique relying on the repeated solution of variational inequality--type problems. A monotone operator theoretic approach is developed which does not resort to Mosco convergence as is often done in connection with existence analysis for QVIs. For the numerical solution of the QVIs a penalty approach combined with a semismooth Newton iteration is proposed. The paper ends with a report on numerical tests involving the $p$-Laplace operator and various types of nonlinear constraints.