4.1.4 The Linear System

  The Newton algorithm requires the solution of the large sparse linearized system (4.1-24) for each iteration. Due to the box integration scheme for the diffusion flux there are only few off-diagonal entries in the system matrix. Our linear system reads (4.1-28).

 

Before solving the linear system we can eliminate equations which degrade the condition of our equation system. Such equations might result from boundary or material interface conditions. Therefore, we split the system matrix into the following structure

where denotes the so-called boundary matrix, the row transformation matrix and gives the interior (segment) matrix. The row transformation matrix is used to specify whether solution variables are eliminated from the system or modified during transformation by the segment matrix and added to the global system matrix . We apply the same transformation to the right-hand-side and get

To assemble the whole system matrix, the following steps have to be done:

• Perform the discretization of the simulation segments to obtain matrix and vector
• Build up the connectivity information to determine the current flow between the segments to fill in the transformation matrix
• Fill in the boundary conditions into the boundary matrices and vector
• Symbolic assembly of the system matrix to evaluate memory requirements
• Assembly of system matrix

For efficient memory management of the system matrix we use the modified compressed sparse row (MCSR) matrix format [Saa90]. Two arrays are used for the storage of large matrices. One array contains the data and the other the pointers to the beginning and ending elements of the matrix rows.

• Pre-elimination
• Scaling
• Pre-conditioning
• Solvers