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Subsections

# 1.4 Components of a Numerical Solution Method

## 1.4.1 Mathematical Model

The starting point of any numerical method is the mathematical model, i.e. the set of equations, usually partial differential or integro-differential equations.

## 1.4.2 Discretization Method

After synthesizing a mathematical model with differential operators, one has to chose a suitable discretization method, i.e. a method of approximating the differential equations by a system of algebraic equations for the variables at some set of discrete locations in space and time. There are various approaches, the most important of which are: finite difference (FD), finite volume (FV) and finite element (FE) methods.

## 1.4.3 Mesh

The discrete locations at which the variables are to be calculated are defined by a mesh which covers the geometric domain on which the problem is to be solved. It divides the solution domain into a finite number of sub-domains (elements, control volumes, etc.)

## 1.4.4 Accuracy

Numerical solutions of physical phenomena are only approximate solutions. In addition to the errors that might be introduced in the course of the development of the solution algorithm, numerical solutions always include three kinds of systematic errors.

• Modeling errors, which are defined as the difference between the actual physical phenomena and the exact solution of the mathematical model;
• Discretization errors, defined as the difference between the exact solution of the equation of the model and the exact solution of that algebraic system of equations obtained by discretizing these equations, and
• Iteration errors, defined as the difference between the iterative and exact solutions of the algebraic systems of equations.

It is important to be aware of the existence of these errors, and even more to try to distinguish them. Various errors may cancel each other, so that sometimes a solution obtained on a coarse mesh pretends to agree better with the experiment than a solution on a finer mesh, which should be more accurate. Therefore meshing has to be dealt with very carefully.    Next: 1.5 Motivation and Outline Up: 1. Introduction Previous: 1.3 IC Fabrication Process

H. Ceric: Numerical Techniques in Modern TCAD