Properties of the matrix

The discrete operator is symmetric in the inner region. This means that $a_{i,j} = a_{j,i}$. Due to the boundary conditions the symmetry of the operator is lost. The reasons for this are:

• When Dirichlet boundary conditions are used the known elements of $x$ should be eliminated in order to keep the matrix symmetric. However this leads to a different dimension of $A, x,$ and $f$, therefore the known elements are not eliminated.
• When dry points occur the derivation of the discrete boundary conditions is already complicated at the interface between wet and dry points. At this moment it is not clear how to discretize these conditions such that the resulting matrix is symmetric.

These difficulties motivate us to use a non-symmetric matrix. This is only a small drawback, because recently good methods have been developed to solve non-symmetric matrix vector systems.


When Neumann conditions are used on all boundaries the resulting matrix is singular. The solution is determined up to a constant. We have to keep this in mind during the construction of the solution procedure.


When Gauss elimination is used to solve equation (5.25), the zero elements in the bend of $A$ become non-zero. This means that the required memory is equal to $2 \times MXC + 2$ vectors. For $MXC$ large, this leads to an unacceptable large amount of memory. Therefore we use an iterative solution method, where the total amount of memory is less than the memory used in the discretization procedure.