Monolithic Overlapping Schwarz Domain Decomposition Methods with GDSW Coarse Spaces for Saddle Point Problems

Monolithic overlapping Schwarz preconditioners for saddle point problems of Stokes, Navier-Stokes, and mixed linear elasticity ty e are presented. For the first time, coarse spaces obtained from the GDSW (Generalized Dryja-Smith-Widlund) approach are used in such a setting. Numerical results of our parallel implementation are presented for several model problems. In particular, cases are considered where the problem cannot or should not b e reduced using local static condensation, e.g., Stokes, Navier-Stokes or mixed elasticity problems with continuous pressure spaces. In the new monolithic preconditioners, the local overlapping problems and the coarse problem are saddle point problems with the same structure as the original problem. Our parallel implementation of these preconditioners is based on the FROSch (Fast and Robust Overlapping Schwarz) library, which is part of the Trilinos package ShyLU. The implementation is algebraic in the sense that the preconditioners can be constructed from the fully assembled stiffness matrix and information about the block structure of the problem. Parallel scalability results for several thousand cores for Stokes, Navier-Stokes, and mixed linear elasticity model problems are reported. Each of the local problems is solved using a direct solver in serial mo de, whereas the coarse problem is solved using a direct solver in serial or MPI-parallel mode or using an MPI-parallel iterative Krylov solve

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

A Continuous/Discontinuous FE Method for the 3D Incompressible Flow Equations

A projection scheme for the numerical solution of the incompressible Navier-Strokes equation is presented. Finite element discontinuous Galerkin (dG) discretization for the velocity in the momentum equations is employed. The incompressibility constraint is enforced by numerically solving the Poisson equation for pressure using a continuous Galerkin (cG) discretization. The main advantage of the method is that is does not require the velocity and pressure approximation spaces to satisfy the usual inf-sup condition, thus equal order finite element approximations for both velocity and pressure can be used. Furthermore, by using cG discretization for the Poisson equation, no auxiliary equations are needed as it is required for dG approximations of second order derivatives. In order to enable large time steps for time marching to steady-state and time evolving problems, implicit scheme is used in connection with high order implicit RK methods. Numerical tests demonstrate that the overall scheme is accurate and computationally efficient

The LifeV library: engineering mathematics beyond the proof of concept

LifeV is a library for the finite element (FE) solution of partial differential equations in one, two, and three dimensions. It is written in C++ and designed to run on diverse parallel architectures, including cloud and high performance computing facilities. In spite of its academic research nature, meaning a library for the development and testing of new methods, one distinguishing feature of LifeV is its use on real world problems and it is intended to provide a tool for many engineering applications. It has been actually used in computational hemodynamics, including cardiac mechanics and fluid-structure interaction problems, in porous media, ice sheets dynamics for both forward and inverse problems. In this paper we give a short overview of the features of LifeV and its coding paradigms on simple problems. The main focus is on the parallel environment which is mainly driven by domain decomposition methods and based on external libraries such as MPI, the Trilinos project, HDF5 and ParMetis. Dedicated to the memory of Fausto Saleri.Comment: Review of the LifeV Finite Element librar

Adaptive time-stepping for incompressible flow. Part II: Navier-Stokes equations

We outline a new class of robust and efficient methods for solving the Navier- Stokes equations. We describe a general solution strategy that has two basic building blocks: an implicit time integrator using a stabilized trapezoid rule with an explicit Adams-Bashforth method for error control, and a robust Krylov subspace solver for the spatially discretized system. We present numerical experiments illustrating the potential of our approach. © 2010 Society for Industrial and Applied Mathematics

Monolithic Multigrid for Magnetohydrodynamics

The magnetohydrodynamics (MHD) equations model a wide range of plasma physics applications and are characterized by a nonlinear system of partial differential equations that strongly couples a charged fluid with the evolution of electromagnetic fields. After discretization and linearization, the resulting system of equations is generally difficult to solve due to the coupling between variables, and the heterogeneous coefficients induced by the linearization process. In this paper, we investigate multigrid preconditioners for this system based on specialized relaxation schemes that properly address the system structure and coupling. Three extensions of Vanka relaxation are proposed and applied to problems with up to 170 million degrees of freedom and fluid and magnetic Reynolds numbers up to 400 for stationary problems and up to 20,000 for time-dependent problems