By switching to a Gauss-Seidel scheme, the lower diagonal block in the matrix can be taken into account. The application of the Gauss-Seidel scheme as a preconditioner is more expensive than the Jacobi scheme by a multiplication of this lower diagonal block only. Jacobi and Gauss-Seidel schemes can be reduced by making the tolerance to which the field equations are solved dependent on the accuracy reached in the outer Krylov iteration. The resulting preconditioner chances at every Krylov iteration step. Numerical experiments have shown that both the block Jacobi and the block Gauss-Seidel schemes diverge when applied as solver.
Further research is necessary to validate the proposed schemes on models of engineering relevance.