Discontinuous Galerkin method for compressible flow and conservation laws

summary:This paper is concerned with the application of the discontinuous Galerkin finite element method to the numerical solution of the compressible Navier-Stokes equations. The attention is paid to the derivation of discontinuous Galerkin finite element schemes and to the investigation of the accuracy of the symmetric as well as nonsymmetric discretization

Nonstandard Finite Element Methods

[no abstract available

Self-Adaptive Methods for PDE

[no abstract available

Robust Numerical Methods for Singularly Perturbed Differential Equations--Supplements

The second edition of the book "Roos, Stynes, Tobiska -- Robust Numerical Methods for Singularly Perturbed Differential Equations" appeared many years ago and was for many years a reliable guide into the world of numerical methods for singularly perturbed problems. Since then many new results came into the game, we present some selected ones and the related sources.Comment: arXiv admin note: text overlap with arXiv:1909.0827

Fluid-structure interaction of compressible flow

The presented work is split into two parts. The first part is devoted to the theory of the discontinuous Galerkin finite element (DGFE) method for the space-time discretization of a nonstationary convection-diffusion initial-boundary value problem with nonlinear convection and linear diffusion. The DGFE method is applied separately in space and time using, in general, different space grids on different time levels and different polynomial degrees p and q in space and time discretization. The main result is the proof of error estimates in L^2(L^2)-norm and in DG-norm formed by the L^2(H^1)-seminorm and penalty terms. The second part of the thesis deals with the realization of fluid-structure interaction problem of the compressible viscous flow with the elastic structure. The time-dependence of the domain occupied by the fluid is treated by the ALE (Arbitrary Lagrangian-Eulerian) method, when the compressible Navier-Stokes equations are formulated in the ALE formulation. The deformation of the elastic body, caused by the aeroelastic forces, is described by the dynamical elasticity equations. Both these systems are coupled by the transmission conditions. For the space discretization of the flow problem the DGFE method is used. The time-discretization is realized by the backward difference formula. The structural problem is discretized by conforming finite element method and the Newmark method. The fluid-structure interaction is realized via weak or strong coupling algorithms. The developed technique is tested by numerical experiments and applied to the simulation of vibrations of vocal folds during phonation onset

A new reconstruction-enhanced discontinuous Galerkin method for time-dependent problems

summary:This work is concerned with the introduction of a new numerical scheme based on the discontinuous Galerkin (DG) method. We propose to follow the methodology of higher order finite volume schemes and introduce a reconstruction operator into the DG scheme. This operator constructs higher order piecewise polynomial reconstructions from the lower order DG scheme. Such a procedure was proposed already in [2] based on heuristic arguments, however we provide a rigorous derivation, which justifies the increased order of accuracy. Numerical experiments are carried out