4. The Assembly Module

Many numerical simulations require the solution of a nonlinear system of partial differential equations. Generally, such a system cannot be solved analytically, and the solution must be calculated by numerical methods. This approach normally consists of three tasks [193]:

1. The domain is partitioned into a finite number of subdomains, in which the solution can be approximated with a desired accuracy.
2. The system of partial differential equations is approximated in each of the subdomains by algebraic equations. The unknowns of the algebraic equations are approximations of the continuous solutions at discrete grid points in the domain. Thus, generally a large system of nonlinear, algebraic equations is obtained with unknowns comprised of approximations of the unknown functions at discrete points.
3. The third task is to derive a solution of the unknowns of the nonlinear algebraic system. In the best case an exact solution of this system can be obtained, which represents a good approximation of the solution of the analytically formulated problem (which cannot be solved exactly). The quality of the approximation depends on the fineness of the partitioning into subdomains as well as on the suitability of the approximating functions for the dependent variables.

This nonlinear problem is usually solved by a damped Newton algorithm (see Section 2.3.1) demanding the solution of a sparse non-symmetric linear equation system at each step. As many simulators, for example MINIMOS-NT, are based on this approach, specific capabilities are required to assemble and solve equation systems. Due to their independence from the other parts of the simulators, these capabilities are frequently incorporated in separate modules.

In this chapter, the assembly module is going to be discussed, subject of the next one is the solver module.