Schwarz type preconditioners for the neutron diffusion equation

• To solve the neutron diffusion equation many linear systems has to be solved using preconditioned Krylov methods.
• Traditional preconditioners based on incomplete factorizations are expensive in terms of memory.
• Domain decomposition preconditioners are studied including substructuring preconditioners and additive Schwarz preconditioners.
• 2D and 3D benchmarks have been studied obtaining better performance results than usual preconditioners.

Domain decomposition is a mature methodology that has been used to accelerate the convergence of partial differential equations. Even if it was devised as a solver by itself, it is usually employed together with Krylov iterative methods improving its rate of convergence, and providing scalability with respect to the size of the problem.In this work, a high order finite element discretization of the neutron diffusion equation is considered. In this problem the preconditioning of large and sparse linear systems arising from a source driven formulation becomes necessary due to the complexity of the problem. On the other hand, preconditioners based on an incomplete factorization are very expensive from the point of view of memory requirements. The acceleration of the neutron diffusion equation is thus studied here by using alternative preconditioners based on domain decomposition techniques inside Schur complement methodology. The study considers substructuring preconditioners, which do not involve overlapping, and additive Schwarz preconditioners, where some overlapping between the subdomains is taken into account.The performance of the different approaches is studied numerically using two-dimensional and three-dimensional problems. It is shown that some of the proposed methodologies outperform incomplete LU factorization for preconditioning as long as the linear system to be solved is large enough, as it occurs for three-dimensional problems. They also outperform classical diagonal Jacobi preconditioners, as long as the number of systems to be solved is large enough in such a way that the overhead of building the preconditioner is less than the improvement in the convergence rate.