Biharmonic Obstacle Problem: Guaranteed and Computable Error Bounds for Approximate Solutions
Biharmonic Obstacle Problem: Guaranteed and Computable Error Bounds for Approximate Solutions
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DOI：
10.1134/s0965542520110032
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发表时间：
2020-11-01
影响因子：
0.7
通讯作者：
Repin, S., I
中科院分区：
文献类型：
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作者：
Apushkinskaya, D. E.;Repin, S., I
研究了一类双调和算子的椭圆型变分不等式的自由边界障碍问题。我们研究了相应变分问题的精确解（极小值）与满足规定边界条件和障碍物规定限制的能量类中的任何函数（逼近）之间的差异的界限。利用广义凸变分问题的一般理论推导出误差恒等式。这个恒等式的一部分表征了函数（近似）与精确解的偏差，而另一部分则是完全计算的值（它只依赖于问题的数据和已知函数）。在真实的计算中，这个恒等式可以用来控制近似解的精度。误差恒等式中使用的精确解的偏差度量包含不同性质的项。其中两个是（直接和对偶变分问题的）精确解和相应的近似之间的差异的范数。另外两个不能作为规范来表示。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），则这些非线性测度为零。误差恒等式对于直接变量的任何可容许（相容）近似都是成立的，但它对对偶变量施加了一些限制。我们表明，这些限制可以被删除，但在这种情况下的身份是由一个不等式。对于正变分问题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显式可计算的优势。几个例子说明了建立的身份和不等式。
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.