For the first time, a nonlinear interface problem on an unbounded domain with nonmonotone set-valued transmission conditions is analyzed. The investigated problem involves a nonlinear monotone partial differential equation in the interior domain and the Laplacian in the exterior domain. Such a scalar interface problem models nonmonotone frictional contact of elastic infinite media. The variational formulation of the interface problem leads to a hemivariational inequality, which lives on the unbounded domain, and so cannot be treated numerically in a direct way. By boundary integral methods the problem is transformed and a novel hemivariational inequality (HVI) is obtained that lives on the interior domain and on the coupling boundary, only. Thus for discretization the coupling of finite elements and boundary elements is the method of choice. In addition smoothing techniques of nondifferentiable optimization are adapted and the nonsmooth part in the HVI is regularized. Thus we reduce the original variational problem to a finite dimensional problem that can be solved by standard optimization tools. We establish not only convergence results for the total approximation procedure, but also an asymptotic error estimate for the regularized HVI.
翻译:首次分析了非线性界面问题。 所调查的问题涉及内域和外域拉平式的非线性单一单色部分方程式。 这种星际界面问题模型没有弹性无限介质摩擦性接触。 界面问题的变式配方导致不均匀的不平等, 它生活在无线域上, 因此无法直接以数字方式处理 。 通过边界整体方法, 问题被改变, 并获得一种新的超异性不平等( HVI), 它只存在于内域和合并边界上。 因此, 将有限元素和边界元素的离散组合是选择的方法。 此外, 平滑的优化技术得到调整, 并且 HVI 中的非光滑部分被固定化 。 因此, 我们把最初的变异性问题降低到可以通过标准优化工具解决的定数维性问题 。 我们不仅为整个近似程序设定了趋同结果, 并且作为常规误差的模拟结果 。