Hydrocarbon production or injection of fluids in the reservoir can produce changes in the rock stresses and in-situ geomechanics, potentially leading to compaction and subsidence with harmful effects in wells, cap-rock, faults, and the surrounding environment as well. In order to tackle these changes and their impact, accurate simulations are essential. The Mortar Finite Element Method (MFEM) has been demonstrated to be a powerful technique in order to formulate a weak continuity condition at the interface of sub-domains in which different meshes, i.e. non-conforming or hybrid, and / or variational approximations are used. This is particularly suitable when coupling different physics on different domains, such as elasticity and poroelasticity, in the context of coupled flow and geomechanics. In this dissertation, popular Domain Decomposition Methods (DDM) are implemented in order to carry large simulations by taking full advantage of current parallel computer architectures. Different solution schemes can be defined depending upon the way information is exchanged between sub-domain interfaces. Three different schemes, i.e. Dirichlet-Neumann (DN), Neumann-Neumann (NN) and MFEM, are tested and the advantages and disadvantages of each of them are identified. As a first contribution, the MFEM is extended to deal with curve interfaces represented by Non-Uniform Rational B-Splines (NURBS) curves and surfaces. The goal is to have a more robust geometrical representation for mortar spaces, which allows gluing non-conforming interfaces on realistic geometries. The resulting mortar saddle-point problem will be decoupled by means of the DN- and NN-DDM. Additionally, a reservoir geometry reconstruction procedure based on NURBS surfaces is presented as well. The technique builds a robust piecewise continuous geometrical representation that can be exploited by MFEM in order to tackle realistic problems, which is a second contribution. Tensor product meshes are usually propagated from the reservoir in a conforming way into its surroundings, which makes non-matching interfaces highly attractive in this case. In the context of reservoir compaction and subsidence estimation, it is common to deal with serial legacy codes for flow. Indeed, major reservoir simulators such as compositional codes lack parallelism. Another issue is the fact that, generally speaking, flow and mechanics domains are different. To overcome this limitation, a serial-parallel approach is proposed in order to couple serial flow codes with our parallel mechanics code by means of iterative coupling. Concrete results in loosely coupling are presented as a third contribution. As a final contribution, the DN-DDM is applied to couple elasticity and plasticity, which seems very promising in order to speed up computations involving poroplasticity. Several examples of coupling of elasticity, poroelasticity, and plasticity ranging from near-wellbore applications to field level subsidence computations help to show that the proposed methodology can handle problems of practical interest. In order to facilitate the implementation of complex workflows, an advanced Python wrapper interface that allows programming capabilities have been implemented. The proposed serial-parallel approach seems to be appropriate to handle geomechanical problems involving different meshes for flow and mechanics as well as coupling parallel mechanistic codes with legacy flow simulators. / text