Abstract

The paper is concerned with estimates of the difference between a given function and the exact solution of an elliptic boundary value problem. Estimates of this type have been derived earlier in terms of the natural energy norm. In this work, an approach is proposed to obtain stronger measures of the deviation and relevant estimates applicable if the exact solution and the approximation have additional regularity (with respect to the order of integrability). These measures include the standard energy norm as a simple special case. A general approach is proposed to construct various measures based on using an auxiliary variational problem. Two classes of measures whose properties are close to those of the \({{L}^{q}}\) and \({{L}^{\infty }}\) norms are studied in more detail. Their properties are established, and explicitly computable two-sided estimates (minorants and majorants) that involve only known functions are constructed.

中文翻译：

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边值问题精确解的偏差估计，其度量要强于能量范数

摘要

本文涉及给定函数与椭圆边界值问题的精确解之间的差异估计。这种类型的估计值是根据自然能量标准在早期得出的。在这项工作中，提出了一种方法，如果精确的解决方案和逼近度具有附加的规律性（关于可积性的阶数），则可以获取更强的偏差度量和适用的相关估计。这些措施包括作为简单特殊情况的标准能源规范。提出了一种基于辅助变分问题构造各种测度的通用方法。两类度量的性质接近\（{{L} ^ {q}} \）和\（{{L} ^ {\ infty}} \）对规范进行了更详细的研究。建立了它们的属性，并构造了仅涉及已知函数的可明确计算的双向估计（米诺和主要）。