BALANCING DOMAIN DECOMPOSITION BY CONSTRAINTS PRECONDITIONERS FOR VIRTUAL ELEMENT DISCRETIZATIONS OF SADDLE-POINT PROBLEMS

The Virtual Element Method (VEM) is a novel family of numerical methods for approximating partial differential equations on general polygonal or polyhedral computational grids. This thesis aims to propose a Balancing Domain Decomposition by Constraints (BDDC) algorithms to compute the solution of the saddle-point linear systems arising from the VEM discretization of the two and three dimensional incompressible Stokes and Oseen equations. These methods are an evolution of Balancing Neumann Neumann preconditioners where, to obtain scalability in the number of subdomains, a coarse space has to ensure global transport of information. In the Stokes model, these methods act as preconditioners making the linear system symmetric and positive definite, this allows to use the conjugate gradient method to accelerate the convergence. For the Oseen model, due to the lack of symmetry the system is still positive definite, but the solution is performed with the generalized minimal residual method (GMRES). In the symmetric framework, we prove the scalability and quasi-optimality of the algorithm and confirm the theoretical findings with several numerical test with parallel computations, including a time analysis for the three dimensional case. We also provide a proof for a Stokes optimal VEM interpolant in the three dimensional framework, needed to complete the analysis of the preconditioner. We extend the BDDC algorithm applied to the non-symmetric saddle point problem that arises from the VEM discretization of the Oseen problem, providing some numerical simulations including scalability and optimality tests in both two and three dimensions. Exploiting the advantages of the VEM methods all our experiments have been performed on different type of meshes with different type of partition techniques. We propose different adaptive technologies to enrich the coarse space both in two and three dimensions, which require the solution of eigenvalue problems on the interface of the subdomains. These techniques are able to control the conditioning of the system and keep it limited below a certain threshold decided a priori. We also provide an heuristic approach that, for some specific cases, efficiently enrich the coarse space overcoming these extra computations. Numerical results with adaptively generated coarse spaces and the new heuristic approach, confirm the method's robustness in the presence of large jumps in the viscosity and with high-order VEM discretizations.