2. Finite Element Method

A good number of practical TCAD problems can be formulated in terms of partial differential equations (PDE) subject to specified boundary conditions. Except in the case of simple and geometrically well-defined problems, closed form analytical solutions are impossible. For complex geometrical shapes, with varying material characteristics and often mixed boundary conditions, only numerical methods allow for a solution.

In this work I discuss the numerical methods for solving PDEs which are used in interconnect and process simulation. The types of equations are very different due to the diversity of the physical phenomena which occur during the processes. There are three widely applied methods for solving of these equations numerically which are based on finite differences, finite boxes, or finite elements.

For the equations considered here the finite element based methods are advantageous. Finite differences are difficult to apply to the non-planar geometries with different accuracy demands in the various parts of the geometry. The finite boxes based method demands explicitely that integral form of the given equation system fulfills some conservation law. For the finite element method these restrictions are not needed.

In this chapter I set overall finite element framework for handling diffusion and electromigration problems.
The basic concepts are motivated and introduced followed by presentation of numerical methods which I later use in Chapter 3 and 4.
At the end of the chapter algorithms for the solving of PDE system, mesh and error control are given.

- 2.1 The Systems of Partial Differential Equations
- 2.2 Rayleigh-Ritz Method
- 2.3 Galerkin's Method
- 2.4 Time Dependent Problems
- 2.5 Finite Element Spaces and Meshes
- 2.6 Newton Methods

- 2.7 Assembling
- 2.8 General Solving Procedure
- 2.9 Time Step and Mesh Control

H. Ceric: Numerical Techniques in Modern TCAD