Gunar Matthies

ParMooNA modernized program package based on mapped finite elements

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 (John and Matthies, 2004): 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 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.

Numerical Study of SUPG and LPS Methods Combined with Higher Order Variational Time Discretization Schemes Applied to Time-Dependent Linear Convection---Diffusion---Reaction Equations

This paper considers the numerical solution of time-dependent linear convection–diffusion–reaction equations. We shall employ combinations of streamline-upwind Petrov–Galerkin and local projection stabilization methods in space with the higher order variational time discretization schemes. In particular, we consider time discretizations by discontinuous Galerkin methods and continuous Galerkin–Petrov methods. Several numerical tests have been performed to assess the accuracy of combinations of spatial and temporal discretization schemes. Furthermore, the dependence of the results on the stabilization parameters of the spatial discretizations are discussed. In addition, the long-time behavior of overshoots and undershoots is studied. The efficient solution of the obtained systems of linear equations by GMRES methods with multigrid preconditioners will be investigated.

Numerical studies of higher order variational time stepping schemes for evolutionary Navier--Stokes equations

We present in this paper numerical studies of higher order variational time stepping schemes combined with finite element methods for simulations of the evolutionary Navier-Stokes equations. In particular, conforming inf-sup stable pairs of finite element spaces for approximating velocity and pressure are used as spatial discretization while continuous Galerkin–Petrov methods (cGP) and discontinuous Galerkin (dG) methods are applied as higher order variational time discretizations. Numerical results for the well-known problem of incompressible flows around a circle will be presented.

A local projection stabilization/continuous Galerkin–Petrov method for incompressible flow problems

Abstract A local projection stabilization (LPS) method in space is considered to approximate the evolutionary Oseen equations. Optimal error bounds with constants independent of the viscosity parameter are obtained in the continuous-in-time case for both the velocity and pressure approximation. In addition, the fully discrete case in combination with higher order continuous Galerkin–Petrov (cGP) methods is studied. Error estimates of order k + 1 are proved, where k denotes the polynomial degree in time, assuming that the convective term is time-independent. Numerical results show that the predicted order is also achieved in the general case of time-dependent convective terms.

Variational Methods for Stable Time Discretization of First-Order Differential Equations

Starting from the well-known discontinuous Galerkin (dG) and continuous Galerkin-Petrov (cGP) time discretization schemes we derive a general class of variational time discretization methods providing the possibility for higher regularity of the numerical solutions. We show that the constructed methods have the same stability properties as dG or cGP, respectively, making them well-suited for the discretization of stiff systems of differential equations. Additionally, we empirically investigate the order of convergence and performance depending on the chosen method.