## Abstract

One might title this chapter “The Challenges of Dimensionality,” or perhaps “Why One-Dimensional Models Aren’t So Bad After All.” The reason is that we address how to solve boundary value problems with more than one spatial variable, and this will require us to consider some unique challenges. To introduce the ideas we will limit the development to two dimensions and consider how to find the function where is the gradient and where where is the Laplacian

*u(x, y)*that satisfies$$
\nabla \cdot (a\nabla u) + b \cdot \nabla u + cu = f,{\text{ }}for{\text{ }}(x,y) \in D,
$$

(6.1)

$$
\nabla \equiv (\frac{\partial }
{{\partial x}},\frac{\partial }
{{\partial y}})
$$

(6.2)

*D*is a bounded domain in the*xy*-plane, as indicated in Figure 6.1. Also, the functions a, b, c, f are smooth with a > 0 and c ≤ 0 on \( \bar D = D \cup \partial D \), where ∂*D*is the boundary of*D*. We will use a Dirichlet boundary condition, which means that the solution is specified around the boundary, and the general form is$$
u = g(x,y),{\text{ }}for{\text{ }}(x,y) \in \partial D,
$$

(6.3)

*g*is given. The particular case of*a*= 1,**b = 0**,*c*= 0 produces Poisson’s equation. If, in addition,*f*= 0, one obtains Laplace’s equation given as$$
\nabla ^2 u = 0,{\text{ }}for{\text{ }}(x,y) \in D,
$$

(6.4)

$$
\nabla ^2 \equiv \frac{{\partial ^2 }}
{{\partial x^2 }} + \frac{{\partial ^2 }}
{{\partial y^2 }}
$$

(6.5)

## Keywords

Matrix Equation Elliptic Problem Steep Descent Iteration Step Conjugate Gradient Method
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media, LLC 2007