In the numerical experiments we discuss test examples and real-life applications with respect to our proposed decomposition methods. We present complex models and their tendency to arrive at ineﬃcient and inexact solutions, and we are forced to search for more detailed and exact solutions for the same problems using simpler equations. In the future, we will propose more solutions that are adequate in their results for multi-physics applications rather than analytical exact solutions, which involve complex equations. Our solutions oﬀer the possibility of mathematically exact proofs of existence and uniqueness, whereas the older computations of greater complexity lack these convergence and existence results. We will defer to mathematical correctness if there is a chance to fulﬁll this in the simpler equations, but we will also describe very complex models and show solvability without proofs of existence and uniqueness. To ﬁnd a balance between simple provable equations and complex calculable equations, we present splitting methods for decoupling complex equations into provable equations. Complex models will be described with more or less understanding of the complexity of the particular systems. Therefore, systematic schemes are used to decouple problems in simpler understandable models and, for the next step, to couple in more complex models. In this way, understanding the part-systems is possible, and the complex model is made at least partially understandable. In this chapter, we introduce various models in solid and ﬂuid mechanics with their physical background, and we discuss mathematical proofs for solutions
For the qualitative characterization of the time decomposition methods, we introduce in the following the benchmark problems.