Finite element methods are conventionally used for solving linear elasticity equations. These methods produce very good results and are widely analyzed from a mathematical point of view to study solid deformations. It seems interesting to have Finite Volume solvers for coupled solid/fluid problems, realistic situations in presence of discontinuities (freezing fronts modeling in wet soils), or even to compute fields better suited to non-conforming meshes. Finite Volume methods are widely used in fluid mechanics. Applied to convection problems, they are well suited to compute solutions with discontinuities and do not require mesh conformity. Moreover, they have the advantage of preserving discrete flows across the interfaces of the mesh. Therefore, we develop and test in this thesis several finite volume methods for solving the elasticity problem. First of all, we implement the LSGR method (Least Squares Gradient Reconstruction), which reconstructs gradients by volume from a weighted least squares formula on neighboring volumes. This method has been successfully tested for unstructured tetrahedral meshes, and shows a first-order convergence rate. Then, we present the Mixed Finite Volume method, based on the conservation of a "penalized" flow across the interfaces. The penalty term imposes a constraint on the type of meshes, and numerical tests are performed in 2D with structured and unstructured quadrangles. Afterwards, we extend the diamond-cell Finite Volume method to the elasticity. This method computes a discrete gradient on sub-volumes related to the interfaces from the interpolation of the solution at vertices. The theoretical convergence is proved under a coercivity condition. The numerical results, achieved in 2d for unstructured meshes, give a second-order convergence rate. Finally, we present the DDFV method (Discrete Duality Finite Volume), which is an extension of the precedent one. This method is based on a correspondence between several meshes in order to construct discrete operators on "discrete duality". We show that the DDFV scheme is a first-order convergent method. The 2d and 3d numerical tests on unstructured meshes show a second-order convergence rate, which is a classical result for this method.
