Abstract

The paper is concerned with an elliptic variational inequality associated with a free boundary obstacle problem for the biharmonic operator. We study the bounds of the difference between the exact solution (minimizer) of the corresponding variational problem and any function (approximation) from the energy class satisfying the prescribed boundary conditions and the restrictions stipulated by the obstacle. Using the general theory developed for a wide class of convex variational problems we deduce the error identity. One part of this identity characterizes the deviation of the function (approximation) from the exact solution, whereas the other is a fully computed value (it depends only on the data of the problem and known functions). In real life computations, this identity can be used to control the accuracy of approximate solutions. The measure of deviation from the exact solution used in the error identity contains terms of different nature. Two of them are the norms of the difference between the exact solutions (of the direct and dual variational problems) and corresponding approximations. Two others are not representable as norms. These are nonlinear measures vanishing if the coincidence set defined by means of an approximate solution satisfies certain conditions (for example, coincides with the exact coincidence set). The error identity is true for any admissible (conforming) approximations of the direct variable, but it imposes some restrictions on the dual variable. We show that these restrictions can be removed, but in this case the identity is replaced by an inequality. For any approximations of the direct and dual variational problems, the latter gives an explicitly computable majorant of the deviation from the exact solution. Several examples illustrating the established identities and inequalities are presented.

中文翻译：

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双谐波障碍问题：近似解的有保证和可计算的误差界

摘要

本文涉及与双调和算子的自由边界障碍问题有关的椭圆变分不等式。我们研究了相应变分问题的精确解（最小化器）与满足预定边界条件的能量类别中的任何函数（逼近）和障碍所规定的限制之间的差的界限。使用针对广泛的凸变分问题开发的一般理论，我们推导出了误差标识。该身份的一部分描述了函数（近似值）与精确解的偏差，而另一部分则是完全计算出的值（它仅取决于问题和已知函数的数据）。在现实生活中的计算中，此标识可用于控制近似解的准确性。与错误标识中使用的精确解的偏差度量包含不同性质的术语。其中两个是（直接和对偶变分问题的）精确解与相应近似之间的差的范数。另外两个不能作为规范来代表。如果通过近似解定义的重合集满足某些条件（例如，与精确重合集重合），则这些非线性度量将消失。对于直接变量的任何允许的（合格的）近似，错误标识都是正确的，但是对双重变量施加了一些限制。我们表明可以消除这些限制，但是在这种情况下，身份由不等式代替。对于直接和对偶变分问题的任何近似，后者给出了与精确解的偏差的可显式计算的主要部分。给出了一些例子，说明了已建立的身份和不平等。