Reliability in finite element method for higher dimensional space problems

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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.