Solution of the obstacle problem by domain decomposition method

Domain decomposition method with non-overlapping subdomains is applied for solving the obstacle problem. An obstacle being located in a known subdomain of the initial domain, partitioning of the domain is made by using this information, so, the information about possible lost of the solution regularity.
Finite element method with quadrature rules and non-matching grids in the subdomains is used to approximate corresponding variational inequality; finer grid is constructed in the subdomain containing the obstacle.
Two iterative methods are constructed to solve finite dimensional problems, they can be viewed as non-linear variants of Douglas–Rachford splitting iterative method. Convergence of the iterative algorithms is proved, their implementation is discussed, and numerical results are applied.