We consider the numerical simulation of the left ventricle of the human heart by a hyperelastic fiber reinforced transversely isotropic model. This is an important model problem for the understanding of the mechanical properties of the human heart but its calculation is very time consuming because the lack of fast, scalable method that is also robust with respect to the model parameters. In this paper, we propose and study a fully implicit overlapping domain decomposition method on unstructured meshes for the discretized system. The algorithm is constructed within the framework of Newton–Krylov methods with an analytically constructed Jacobian. We show numerically that the algorithm is highly parallel and robust with respect to the material parameters, the large deformation, the fiber reinforcement, and the geometry of the patient-specific left ventricle. Numerical experiments show that the algorithm scales well on a supercomputer with more than 8000 processor cores.