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

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

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

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

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

where lin() is defined in the following way:

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