4. 2 Element-Wise Assembly

An alternative to the line-wise assembly as shown above is the assembly via small local sub-matrices or stencil matrices. The major advantage of this method over the line-wise assembly is that normally only one incidence relation is required. The method of element-wise assembly typically used for finite element and finite volume simulation tools.

In contrast to the line-wise assembly, element-wise assembly considers the influence of the topological elements of highest order used in the topological neighborhood, mostly cells, on the final equation system. If used for the solution of a differential equation with one solution function, each cell influences a set of $n \times n$ matrix entries where $n$ is the number of incident of the element.