Efficient and Robust Solvers based on Multigrid Methods

Comparison of Preconditioners

Geometric multigrid vs. LSC
Naveed Ahmed, Clemens Bartsch, Volker John, Ulrich Wilbrandt An Assessment of Some Solvers for Saddle Point Problems Emerging from the Incompressible Navier-Stokes Equations, Comput. Methods Appl. Mech. Engrg. 331, 492 - 513, 2018
Efficient incompressible flow simulations, using inf-sup stable pairs of finite element spaces, require the application of efficient solvers for the arising linear saddle point problems. This paper presents an assessment of different solvers: the sparse direct solver UMFPACK, the flexible GMRES (FGMRES) method with different coupled multigrid preconditioners, and FGMRES with Least Squares Commutator (LSC) preconditioners. The assessment is performed for steady-state and time-dependent flows around cylinders in 2d and 3d. Several pairs of inf-sup stable finite element spaces with second order velocity and first order pressure are used. It turns out that for the steady-state problems often FGMRES with an appropriate multigrid preconditioner was the most efficient method on finer grids. For the time-dependent problems, FGMRES with LSC preconditioners that use an inexact iterative solution of the velocity subproblem worked best for smaller time steps.

Parallel Geometric Multigrid Method

Parallel geometric multigrid vs. PETSc routines
Ulrich Wilbrandt, Clemens Bartsch, Naveed Ahmed, Najib Alia, Felix Anker, Laura Blank, Alfonso Caiazzo Sashikumaar Ganesan, Swetlana Giere, Gunar Matthies, Raviteja Meesala, Abdus Shamim, Jagannath Venkatesan, Volker John ParMooN - a modernized program package based on mapped finite elements, Computers and Mathematics with Applications 74, 74 - 88, 2017
{\sc ParMooN} is a program package for the numerical solution of elliptic and parabolic partial differential equations. It inherits the distinct features of its predecessor {\sc MooNMD} \cite{JM04}: strict decoupling of geometry and finite element spaces, implementation of mapped finite elements as their definition can be found in textbooks, and a geometric multigrid preconditioner with the option to use different finite element spaces on different levels of the multigrid hierarchy. After having presented some thoughts about in-house research codes, this paper focuses on aspects of the parallelization for a distributed memory environment, which is the main novelty of {\sc ParMooN}. Numerical studies, performed on compute servers, assess the efficiency of the parallelized geometric multigrid preconditioner in comparison with some parallel solvers that are available in the library {\sc PETSc}. The results of these studies give a first indication whether the cumbersome implementation of the parallelized geometric multigrid method was worthwhile or not.

Multigrid Methods for Higher Order Finite Elements

Time-dependent 3D flow through a channel around a cylinder
V. John "On the Efficiency of Linearization Schemes and Coupled Multigrid Methods in the Simulation of a 3D Flow Around a Cylinder",, Int. J. Numer. Meth. Fluids 50, 845 - 862, 2006
This paper studies the efficiency of two ways to treat the non-linear convective term in the time-dependent incompressible Navier-Stokes equations and of two multigrid approaches for solving the arising linear algebraic saddle point problems. The Navier-Stokes equations are discretized by a second order implicit time stepping scheme and by inf-sup stable, higher order finite elements in space. The numerical studies are performed at a 3D flow around a cylinder.
Analysis of the multiple discretization multigrid method for symmetric positive definite saddle point problems
V. John, P. Knobloch, G. Matthies and L. Tobiska , "Non-nested multi-level solvers for finite element discretizations of mixed problems" , Computing 68, 313 - 341, 2002
We consider a general framework for analysing the convergence of multi-grid solvers applied to finite element discretisations of mixed problems, both of conforming and nonconforming type. As a basic new feature, our approach allows to use different finite element discretisations on each level of the multi-grid hierarchy. Thus, in our multi-level approach, accurate higher order finite element discretisations can be combined with fast multi-level solvers based on lower order (nonconforming) finite element discretisations. This leads to the design of efficient multi-level solvers for higher order finite element discretisations.
3D flow through a channel around a cylinder, comprehensive study
V. John "Higher order finite element methods and multigrid solvers in a benchmark problem for the 3D Navier-Stokes equations",, Int. J. Num. Meth. Fluids 40, 775 - 798, 2002
This paper presents a numerical study of the 3D flow around a cylinder which was defined as a benchmark problem for the steady state Navier-Stokes equations within the DFG high priority research program Flow Simulation with High--Performance Computers by Sch\"afer and Turek (1996). The first part of the study is a comparison of several finite element discretizations with respect to the accuracy of the computed benchmark parameters. It turns out that boundary fitted higher order finite element methods are in general most accurate. Our numerical study improves the hitherto existing reference values for the benchmark parameters considerably. The second part of the study deals with efficient and robust solvers for the discrete saddle point problems. All considered solvers are based on coupled multigrid methods. The flexible GMRES method with a multiple discretization multigrid methods proves to be the best solver.
2D flow through a channel around a cylinder, brief study
V. John and G. Matthies, " Higher Order Finite Element Discretizations in a Benchmark Problem for Incompressible Flows", Int. J. Num. Meth. Fluids 37, 885 - 903, 2001
We present a numerical study of several finite element discretizations applied to a benchmark problem for the 2d steady state incompressible Navier-Stokes equations defined in Sch\"afer and Turek (1996). The discretizations are compared with respect to the accuracy of the computed benchmark parameters. Higher order isoparametric finite element discretizations turned out to be by far most accurate. The discrete systems obtained with higher order discretizations are solved with a modified coupled multigrid method whose behaviour within the benchmark problem is also studied numerically.

Multigrid Methods for Lowest Order Non-Conforming Finite Elements on Parallel Computers

Comprehensive study of smoothers in coupled multigrid methods for saddle point problems in 2D
V. John, L. Tobiska, "Numerical Performance of Smoothers in Coupled Multigrid Methods for the Parallel Solution of the Incompressible Navier--Stokes Equations", Int. J. Num. Meth. Fluids 33, 453 - 473, 2000
In recent benchmark computations, coupled multigrid methods have been proven as efficient solvers for the incompressible Navier-Stokes equations. We present a numerical study of two classes of smoothers in the framework of coupled multigrid methods. The class of Vanka--type smoothers is characterized by the solution of small local linear systems of equations in a Gauss-Seidel manner in each smoothing step whereas the Brass-Sarazin-type smoothers solve a large global saddle point problem. The behaviour of these smoothers with respect to computing times and parallel overhead is studied for two--dimensional steady state and time dependent Navier-Stokes equations.
Analysis of a multigrid method for non-conforming finite elements in 2D
V. John, L. Tobiska, "A Coupled Multigrid Method for Nonconforming Finite Element Discretizations of the 2d-Stokes Equation", Computing 64, 307 - 321, 2000
This paper investigates a multigrid method for the solution of the saddle point formulation of the discrete Stokes equation obtained with inf-sup stable nonconforming finite elements of lowest order. A smoother proposed by Braess and Sarazin (1997) is used and $L^2$-projection as well as simple averaging are considered as prolongation. The W-cycle convergence in the $L^2$-norm of the velocity with a rate independently of the level and linearly decreasing with increasing number of smoothing steps is proven. Numerical tests confirm the theoretically predicted results.
Numerical study of strongly and weakly coupled multigrid methods for saddle point problems
V. John, "A Comparison of Parallel Solvers for the Incompressible Navier-Stokes Equations", Comput. Visual. Sci. 1 (4), 193 - 200, 1999
In this paper we compare coupled multigrid methods and some pressure correction schemes (operator splitting schemes) for the solution of the steady state and time dependent incompressible Navier--Stokes equations. We consider pressure correction schemes with multigrid as well as single grid methods for the solution of the Schur complement problem for the pressure. The numerical tests have been carried out on benchmark problems using a MIMD parallel computer. They show the superiority of the coupled multigrid methods for the considered class of problems.