At the beginning of this thesis our main goal was to define fundamental equations and relations from Partial Differential Equations which we will use here. Essential classification of domain decomposition techniques we mentioned here, and we specified which ones are going to be in our focus. We studied direct solution and checked relations between matrix A and its Schur complement S , and analized those when A and S were symmetric positive definite matrices. Regarding the Schur complement we concentrated on vertex-based partitions. Multiplicative Schwarz procedure on each subdomain sequentially gets solution while using boundary conditions for current step which we get from last value of the solution. Additive Schwarz procedure doesn’t update components in each subdomain until a whole cycle of updates through all domains are completed. When A is symmetric positive definite matrix we checked what are convergence conditions for Schwarz alternating methods. Induced preconditioners are based on Schur complement and it basically uses ILU factorization of original matrix A, while preconditioning of vertex-based Schur complements is based on using LU factorization of local A matrix. At the end we checked full matrix method, these methods are based on iterations on the original system, here we also use approximating interface values to get preconditioners of A while for Schur complement we use LU factorization.