Solving Phase-Field Equations with Finite Elements

Abstract

The finite element method, FEM, and sometimes also called finite element analysis, FEA, was originally developed in the aircraft industry in 1960s [1, 2]. Therefore, it is a very mature algorithm and widely used in engineering and science as a general numerical approach for the solution of PDEs subject to known boundary and initial conditions. The use of piecewise continuous functions over subregions of domain to approximate the unknown function was first introduced by Courant [3]. This approach was later formalized [4, 5] and term finite elements for these subregions was introduced by Clough [6]. Therefore, similar to finite difference technique the FEM is also local in nature. However, FEM has superior and unique characteristics to describe very complex geometries and boundaries of domains. There are plenty of textbook and online materials covering both theoretical and practical aspects of FEM or FEA and some of them are listed in the reference section.

The original version of this chapter was revised. An erratum to this chapter can be found at DOI 10.1007/978-3-319-41196-5_9