The discretization of partial differential equations using finite element or finite volume methods often leads to high-dimensional equation systems that are too large to still be solved efficiently on a single computer. In this seminar we will explore different approaches to overcome this limitation:
Domain decomposition methods use decompositions of the computational domain into smaller subdomains, on which easily solvable sub-problems are formulated. These coupled sub-problems can then be solved in parallel on large compute clusters.
When the problem has multiple spatial or temporal scales, numerical multiscale methods can be used to formulate local cell problems which extract the necessary fine-scale information to then build an effective global macro-scale problem of small dimensions.
Often, the same problem has to be solved repeatedly for varying physical or geometrical parameters. In this case, model order reduction techniques can be used to build a low-dimensional surrogate model, which can be solved quickly for varying parameter values while retaining control over the approximation error.