# Fast Methods

The simulation of partial differential equations is a complex computational task. Discretization by finite elements or similar methods leads to a large scale system of equations that has to be solved. General algorithms for their solution are usually too slow and consume too much memory. Thus, the development of specialized solvers for this class of problems is important to make their solution possible with reasonable computational effort.

# Fast Methods

**Coordination:**Prof. Dr. Mario Bebendorf

**Contact:**Prof. Dr. Mario Bebendorf

**Represented in MODUS since**September 2016

**Members with experience in this area:**

**The method "Fast Methods":**

## What is this?

The simulation of partial differential equations is a complex computational task. Discretization by finite elements or similar methods leads to a large scale system of equations that has to be solved. General algorithms for their solution are usually too slow and consume too much memory. Thus, the development of specialized solvers for this class of problems is important to make their solution possible with reasonable computational effort.

## What is it good for?

Partial differential equations arise in many areas of physics, such as thermodynamics, solid and fluid mechanics, or electro-magnetism. In general, phenomena with distributed physical quantities (fields) are modelled with partial differential equations. Their qualitative and quantitative understanding is one of the important issues in mathematics and physics.

An accurate computation of solutions of these equations, however, is a computationally complex and expensive task. It is necessary to represent the given geometry and the desired field in the computer. Consequently the data to be processed is very large scale. In fact, after discretization by finite elements the computational task is to solve a very large linear or non-linear system of equations.

Classical methods for this task have a complexity of O(N^3), where N is the number of degrees of freedom. This would limit the resolution and thus the accuracy of the numerical method to only moderate size. Thus a lot of research has been conducted to develop fast solvers for these problems who have the complexity O(N), opening the way to much more accurate simulation algorithms. The key to this high efficiency is the use of problem inherent structure that is present in partial differential equations.