Extension of the parallel Sparse Matrix Vector Product (SpMV) for the implicit coupling of PDEs on non-matching meshes

The Sparse Matrix Vector Product (SpMV) is one of the main operations of iterative solvers, and, in a parallel context, it is also the siege of point-to-point communications between the neighboring MPI processes. The parallel SpMV is built in such a way that it gives, up to round off errors, the same result as its sequential counterpart. In this regards, nodes on the interfaces (or halo nodes if halo are considered) are duplicated nodes of the same original mesh. It is therefore limited to matching meshes. In this work, we generalize the parallel SpMV to glue the solution of non-matching (non-conforming) meshes through the introduction of transmission matrices. This extension of the SpMV thus enables the implicit and parallel solution of partial differential equations on non-matching meshes, as well as the implicit coupling of multiphysics problems, such as fluid-structure interactions. The proposed method is developed similarly to classical parallelization techniques and can therefore be implemented by modifying few subroutines of an already MPI-based code. According to the proposed framework, the classical parallelization technique appears as a particular case of this general setup.