Additive domain decomposition operator splittings -- convergence analyses in a dissipative framework

We analyze temporal approximation schemes based on overlapping domain decompositions. As such schemes enable computations on parallel and distributed hardware, they are commonly used when integrating large-scale parabolic systems. Our analysis is conducted by first casting the domain decomposition procedure into a variational framework based on weighted Sobolev spaces. The time integration of a parabolic system can then be interpreted as an operator splitting scheme applied to an abstract evolution equation governed by a maximal dissipative vector field. By utilizing this abstract setting, we derive an optimal temporal error analysis for the two most common choices of domain decomposition based integrators. Namely, alternating direction implicit schemes and additive splitting schemes of first and second order. For the standard first-order additive splitting scheme we also extend the error analysis to semilinear evolution equations, which may only have mild solutions.Comment: Please refer to the published article for the final version which also contains numerical experiments. Version 3 and 4: Only comments added. Version 2, page 2: Clarified statement on stability issues for ADI schemes with more than two operator

Hybrid PDE solver for data-driven problems and modern branching

The numerical solution of large-scale PDEs, such as those occurring in data-driven applications, unavoidably require powerful parallel computers and tailored parallel algorithms to make the best possible use of them. In fact, considerations about the parallelization and scalability of realistic problems are often critical enough to warrant acknowledgement in the modelling phase. The purpose of this paper is to spread awareness of the Probabilistic Domain Decomposition (PDD) method, a fresh approach to the parallelization of PDEs with excellent scalability properties. The idea exploits the stochastic representation of the PDE and its approximation via Monte Carlo in combination with deterministic high-performance PDE solvers. We describe the ingredients of PDD and its applicability in the scope of data science. In particular, we highlight recent advances in stochastic representations for nonlinear PDEs using branching diffusions, which have significantly broadened the scope of PDD. We envision this work as a dictionary giving large-scale PDE practitioners references on the very latest algorithms and techniques of a non-standard, yet highly parallelizable, methodology at the interface of deterministic and probabilistic numerical methods. We close this work with an invitation to the fully nonlinear case and open research questions.Comment: 23 pages, 7 figures; Final SMUR version; To appear in the European Journal of Applied Mathematics (EJAM

Parallel Overlapping Schwarz Preconditioners for Incompressible Fluid Flow and Fluid-Structure Interaction Problems

Efficient methods for the approximation of solutions to incompressible fluid flow and fluid-structure interaction problems are presented. In particular, partial differential equations (PDEs) are derived from basic conservation principles. First, the incompressible Navier-Stokes equations for Newtonian fluids are introduced. This is followed by a consideration of solid mechanical problems. Both, the fluid equations and the equation for solid problems are then coupled and a fluid-structure interaction problem is constructed. Furthermore, a discretization by the finite element method for weak formulations of these problems is described. This spatial discretization of variables is followed by a discretization of the remaining time-dependent parts. An implementation of the discretizations and problems in a parallel C++ software environment is described. This implementation is based on the software package Trilinos. The parallel execution of a program is the essence of High Performance Computing (HPC). HPC clusters are, in general, machines with several tens of thousands of cores. The fastest current machine, as of the TOP500 list from November 2019, has over 2.4 million cores, while the largest machine possesses over 10 million cores. To achieve sufficient accuracy of the approximate solutions, a fine spatial discretization must be used. In particular, fine spatial discretizations lead to systems with large sparse matrices that have to be solved. Iterative preconditioned Krylov methods are among the most widely used and efficient solution strategies for these systems. Robust and efficient preconditioners which possess good scaling behavior for a parallel execution on several thousand cores are the main component. In this thesis, the focus is on parallel algebraic preconditioners for fluid and fluid-structure interaction problems. Therefore, monolithic overlapping Schwarz preconditioners for saddle point problems of Stokes and Navier-Stokes problems are presented. Monolithic preconditioners for incompressible fluid flow problems can significantly improve the convergence speed compared to preconditioners based on block factorizations. In order to obtain numerically scalable algorithms, coarse spaces obtained from the Generalized Dryja-Smith-Widlund (GDSW) and the Reduced dimension GDSW (RGDSW) approach are used. These coarse spaces can be constructed in an essentially algebraic way. Numerical results of the parallel implementation are presented for various incompressible fluid flow problems. Good scalability for up to 11 979 MPI ranks, which corresponds to the largest problem configuration fitting on the employed supercomputer, were achieved. A comparison of these monolithic approaches and commonly used block preconditioners with respect to time-to-solution is made. Similarly, the most efficient construction of two-level overlapping Schwarz preconditioners with GDSW and RGDSW coarse spaces for solid problems is reported. These techniques are then combined to efficiently solve fully coupled monolithic fluid-strucuture interaction problems