ParMooN—A modernized program package based on mapped finite elements

Abstract

ParMooN is a program package for the numerical solution of elliptic and parabolic partial differential equations. It inherits the distinct features of its predecessor MooNMD [28]: 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 hierar- * Corresponding author.Email addresses: ulrich.wilbrandt@wias-berlin.de (Ulrich Wilbrandt), clemens.bartsch@wias-berlin.de (Clemens Bartsch), naveed.ahmed@wias-berlin.de (Naveed Ahmed), najib.alia@wias-berlin.de (Najib Alia), felix.anker@wias-berlin.de (Felix Anker), laura.blank@wias-berlin.de (Laura Blank), alfonso.caiazzo@wias-berlin.de (Alfonso Caiazzo), sashi@cds.iisc.ac.in (Sashikumaar Ganesan), swetlana.giere@wias-berlin.de (Swetlana Giere), gunar.matthies@tu-dresden.de (Gunar Matthies), raviteja@cmg.cds.iisc.ac.in (Raviteja Meesala), shamim@cmg.cds.iisc.ac.in (Abdus Shamim), jagan@cmg.cds.iisc.ac.in (Jagannath Venkatesan), volker.john@wias-berlin.de (Volker John)1 The work of Najib Alia has been supported by a funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No. 675715 (MIMESIS).2 The work of Felix Anker has been supported by grant Jo329/10-2 within the DFG priority programme 1679: Dynamic simulation of interconnected solids processes.3 The work of Sashikumaar Ganesan has partially been supported by the Naval Research Board, DRDO, India through the grant NRB/4003/PG/368. 4 The work of Volker John has partially been supported by grant Jo329/10-2 within the DFG priority programme 1679: Dynamic simulation of interconnected solids processes. chy. 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 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 PETSc. The results of these studies give a first indication whether the cumbersome implementation of the parallelized geometric multigrid method was worthwhile or not.

Finite elements for scalar convection-dominated equations and incompressible flow problems: a never ending story?

“…The efficiency of computing this solution is the key for the efficiency of the whole simulation. Recent numerical studies in [140,2] show that socalled coupled multigrid methods with Vanka-type smoothers [137] are often much more efficient than certain one-grid solvers, in particular on fine grids. However, the geometry in applications might be so complex that only one grid is available and multigrid methods cannot be applied.…”

“…However, the geometry in applications might be so complex that only one grid is available and multigrid methods cannot be applied. Among the one-grid solvers, sparse direct solvers were particularly inefficient for three-dimensional problems in [140,2]. The numerical studies in [140,2] considered as one-grid solver flexible GMRES (FGMRES) [128] with preconditioners of least squares commutator (LSC) type [68,69].…”

“…Among the one-grid solvers, sparse direct solvers were particularly inefficient for three-dimensional problems in [140,2]. The numerical studies in [140,2] considered as one-grid solver flexible GMRES (FGMRES) [128] with preconditioners of least squares commutator (LSC) type [68,69]. In three dimensions, this approach worked on the one hand much more efficient than the used sparse direct solver but on the other hand much less efficient than FGMRES with coupled multigrid preconditioners.…”

Numerical comparisons of finite element stabilized methods for a 2D vortex dynamics simulation at high Reynolds number

“…For the one-level variant of LPS method that needs enriched FE spaces for velocities, we used mapped FE spaces [31], where the enriched space on the reference cell K = (−1, 1) 2 is defined by…”

Analysis of a stabilized penalty-free Nitsche method for the Brinkman, Stokes, and Darcy problems

“…In particular, line colors will denote different values of α (GLS stabilization), line marker will refer to δ (Grad-Div stabilization) and line style will be related to the value of the characteristic length L 0 . The numerical solutions have been computed using the finite element library ParMooN [33]. Figure 2.…”