arXiv subjects

Volker John

Publications and source records attributed to Volker John.

21 records · Page 2Linked to original sources

arXiv2017

ParMooN - a modernized program package based on mapped finite elements

{\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.

math.NA↗

arXiv2017

Analysis of the grad-div stabilization for the time-dependent Navier-Stokes equations with inf-sup stable finite elements

This paper studies inf-sup stable finite element discretizations of the evolutionary Navier--Stokes equations with a grad-div type stabilization. The analysis covers both the case in which the solution is assumed to be smooth and consequently has to satisfy nonlocal compatibility conditions as well as the practically relevant situation in which the nonlocal compatibility conditions are not satisfied. The constants in the error bounds obtained do not depend on negative powers of the viscosity. Taking into account the loss of regularity suffered by the solution of the Navier--Stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data, error bounds of order $\mathcal O(h^2)$ in space are proved. The analysis is optimal for quadratic/linear inf-sup stable pairs of finite elements. We also consider the analysis of the fully discrete case with the backward Euler method as time integrator.

math.NA↗

arXiv2016

An object oriented parallel finite element scheme for computations of PDEs: Design and implementation

Parallel finite element algorithms based on object-oriented concepts are presented. Moreover, the design and implementation of a data structure proposed are utilized in realizing a parallel geometric multigrid method. The ParFEMapper and the ParFECommunicator are the key components of the data structure in the proposed parallel scheme. These classes are constructed based on the type of finite elements (continuous or nonconforming or discontinuous) used. The proposed solver is compared with the open source direct solvers, MUMPS and PasTiX. Further, the performance of the parallel multigrid solver is analyzed up to 1080 processors. The solver shows a very good speedup up to 960 processors and the problem size has to be increased in order to maintain the good speedup when the number of processors are increased further. As a result, the parallel solver is able to handle large scale problems on massively parallel supercomputers. The proposed parallel finite element algorithms and multigrid solver are implemented in our in-house package ParMooN.

cs.MS↗