Reliability in finite element method for higher dimensional space problems

数学建模在科学和工程中起着至关重要的作用。昂贵和耗时的实验被计算分析所取代。在工业中，商业代码被广泛使用。它们是灵活的，可以根据解决感兴趣的特定问题进行调整。解决有数万或数十万个未知数的大问题成了例行公事。计算分析的目标是预测工程和物理现实的行为，通常在成本和时间的约束下。如今，人力和时间成本比计算机成本更重要。这一趋势将持续到未来。计算结果与实际情况的一致性与两个因素有关，即问题的数学表述和数值解的精度。必须在分析目的的背景下理解准确性。一个不适当的标准中的一个小误差并不一定意味着计算结果可以用于实际目的。用不同的方法分析同一工程问题有时会导致显著不同的结果。这可能是由各种因素造成的，例如，使用了不同的数学公式中的模型，或者数值解不能很好地逼近感兴趣的数据。当使用具有后验误差估计的各种现代自适应编码时，尤其可能发生这种情况。显然，了解造成这种差异的原因是非常重要的。尽管计算机是主要的工具，但工程上的决策仍然是由人而不是计算机做出的。有必要认识到，计算机总是提供数据、彩色图表或电影，无论正确与否。因此，了解建模的基本数学背景是非常重要的。我研究的主要目的是为创造性地评估计算机提供的数据是否可靠和值得信赖，作为关键工程决策的基础做出重大贡献。

Mathematical modeling plays an essential role in science and engineering. Costly and time-consuming experiments are replaced by computational analysis. In industry, commercial codes are widely used. They are flexible and can be adjusted for solving specific problems of interest. Solving large problems with tens or hundreds of thousands of unknowns becomes routine. The objective of computational analysis is to predict the behavior of the engineering and physical reality usually within the constraints of cost and time. Today, human and time costs are more important than computer costs. This trend will continue into the future.Agreement between computational results and reality is related to two factors, namely the mathematical formulation of the problems and the accuracy of the numerical solution. The accuracy has to be understood in the context of the aim of the analysis. A small error in an inappropriate norm does not necessarily mean that the computed results are usable for practical purposes. Analyzing the same engineering problem by different methods could sometimes lead to results that differ significantly. This could be caused by various factors, e.g., different models in mathematical formulations were used or the numerical solution does not approximate the data of interest well. This can happen especially when various modern adaptive codes with a posteriori error estimation are used. Obviously, to understand the reasons for such discrepancies is very important. Decisions in engineering are still made by humans and not by computers, although computers are the main tools. It is necessary to realize that the computer always provides data, color graphs or movies, correct or incorrect. Hence, to understand the basic mathematical background of the modeling is of major importance. The main objective of my research is to contribute significantly to the creative assessment of whether the numbers provided by computers are reliable and trustworthy as the basis for crucial engineering decisions.