4. 1. 4 Linear Expressions and Functional Description

In the following sections the linear dependence of equations on different solution variables $ x_i$ is discussed. Typically, a solution variable is defined as a quantity on the underlying cell complex. Furthermore, each quantity value that is a solution value requires to be assigned a definite position $ i$ in the solution vector.

In order to determine the position of the quantity associated with a given topological element $ \mathbf{v}$ , an index function $ i(\mathbf{v})$ is introduced. If more solution quantities are required, the function defermining the position of the solution within the vector can be obtained by different index functions ($ i_n$ and $ i_{\psi}$ for the quantities $ n$ and $ \psi$ ).

In the following the residual expressions of discretized differential equations are formulated with linearized expressions. A residual expression is formulated and defines a dependence between single quantity values

$\displaystyle R = f(q(\mathbf{v}_1), q(\mathbf{v}_2)) (= 0) \; .$ (4.11)

Initially the quantity $ q$ may have any value and can not be neglected. The equation system may depend on the quantity $ q$ which is evaluated on many different topological elements. For each of these elements an index function $ i$ is available that assigns each topological element a position in the solution vector. The formulation of the residual equation (4.11) can be written as follows:

$\displaystyle R(\mathbf{v}) = q(\mathbf{v}) + 1 \cdot x_{(i(\mathbf{v}))}\; .$ (4.12)

This expression can be written using the lin() function as

$\displaystyle R(\mathbf{v}) = [\mathrm{lin}(q, i)](\mathbf{v})\; \; ,$ (4.13)

where lin() is defined in the following way:

$\displaystyle Q := \mathrm{lin}(q, j) = q + x_j = [ \underbrace{ \ldots , 1, \ldots }_\mathrm{j-th\;position}; q] \; .$ (4.14)

In the following examples, the residual expressions $ q$ are replaced by their linearized analoga $ Q$ , which implies that each quantity is added an increment $ x_j$ , where $ j$ is the position, given by the index function $ i$ . This function $ i$ represents the position of the matrix column which is relevant for the quantity $ q$ on the given element. If the residual equations are given in this manner, the solution consists of a vector of solution variables $ x_j$ which are added to the quantities $ q$ in order to obtain the final solution.

Michael 2008-01-16