Groundbreaking Ideas in Numerics and their Effect

Groundbreaking ideas in numerics are connected with great names, such as Newton, Euler, Gauß etc. In numerics of partial differential equations, the major ideas go back to the last two centuries, two of them are central. The first is the finite element method which goes back to K. H. Schellbach, 1851, the second is the iterative solution of large sparse algebraic systems going essentially back to Gauß, 1823.

Numerics were boosted by the introduction of computers and their exploding computational power. Without numerics, however, computers would be mere data shuffling machines, not problem solvers. With increasing computational power, complexity became more and more the dominating issue of numerical computations. After Gauß‘s invention of iterative solvers for linear systems of equations a second idea was necessary to overcome the complexity problem. This second step was the multigrid idea, first described by Brakhage, 1960, and thoroughly understood and mathematically analyzed by Wolfgang Hackbusch in 1976 and the following years.

This multigrid idea can be viewed as a special variant of the multiscale character of the physical world. The analytical investigation thereof has been initiated by Einstein, 1905, and continued by numerous scientists throughout the 20th century to date. The multigrid idea, however, was conceived and introduced completely independent of these physical techniques, merely by studying the properties of the corresponding methods and equations.

Nowadays, computing plays a central and more and more decisive role in industrial development and design. Thus, multigrid solvers are crucial components of most simulation codes used in industrial simulation. Simulation in industry has become a big business, which finally is possible only due to the groundbreaking idea of multigrid methods.

Fast and robust multigrid methods allow a new dimension of numerics for the realistic approximation and computation of Physics. This gives a new boost to applied research, in particular in the life sciences. In the talk, we first introduce the basic ideas, then we present applications and show results from several application areas.