This thesis is conducted in the field of numerical analysis which is part of applied mathematics. More precisely we study some methods to solve certain linear systems. The linear systems that we consider are derived from discretizations of partial differential equations. Such linear systems often inherit the properties of the underlying partial differential equation. For example the corresponding matrix of the linear system is sparse. This property motivates the use of iterative methods for the solution technique of such linear systems, since the multiplication of a sparse matrix with a vector is computationally cheap. In this thesis we propose one such iterative method and prove rigorously its advantage over other iterative methods.