Approximation of functions

Authors:Hans Petter Langtangen Date:2016

PRELIMINARY VERSION

The finite element method is a powerful tool for solving partial differential equations. The method can easily deal with complex geometries and higher-order approximations of the solution. Below is a two-dimensional domain with a non-trivial geometry.

The idea of the finite element method is to divide the domain into triangles (elements) and seek a polynomial approximations to the unknown functions on each triangle. The method glues these piecewise approximations together to find a global solution. Linear and quadratic polynomials over the triangles are particularly popular, because of their mathematical simplicity, but higher-degree polynomials are advantageous to create very computationally efficient methods. The reason for using triangles is that they can easily approximate geometrically complicated domains, but quadrilateral elements and boxes in 3D are also widely used.