Abstract In this note we consider a parabolic evolution equation in a polygonal space-time cylinder. We show, that the elliptic part is given by a m-accretive mapping from L q (Ω) → L q (Ω). Therefore we can apply the theory of nonlinear semigroups in Banach spaces in order to get regularity results in time and space. The second part of the paper deals with the numerical solution of the problem. It is dedicated to the application of the space-time discontinuous Galerkin method (STDGM). It means that both in space as well as in time discontinuous piecewise polynomial approximations of the solution are used. We concentrate to the theoretical analysis of the error estimation.

This study deals with the development of an accurate, efficient and robust method for the numerical solution of the interaction of compressible flow and nonlinear dynamic elasticity. This problem requires the reliable solution of flow in time-dependent domains and the solution of deformations of elastic bodies formed by several materials with complicated geometry depending on time. In this paper, the fluid–structure interaction (FSI) problem is solved numerically by the space-time discontinuous Galerkin method (STDGM). In the case of compressible flow, we use the compressible Navier–Stokes equations formulated by the arbitrary Lagrangian–Eulerian (ALE) method. The elasticity problem uses the non-stationary formulation of the dynamic system using the St. Venant–Kirchhoff and neo-Hookean models. The STDGM for the nonlinear elasticity is tested on the Hron–Turek benchmark. The main novelty of the study is the numerical simulation of the nonlinear vocal fold vibrations excited by the compressible airflow coming from the trachea to the simplified model of the vocal tract. The computations show that the nonlinear elasticity model of the vocal folds is needed in order to obtain substantially higher accuracy of the computed vocal folds deformation than for the linear elasticity model. Moreover, the numerical simulations showed that the differences between the two considered nonlinear material models are very small.

The paper deals with the discontinuous Galerkin method (DGM) for the solution of compressible Navier-Stokes equations in the ALE form in time-dependent domains combined with the solution of linear and nonlinear dynamic elasticity. The developed methods are oriented to fluid-structure interaction (FSI), particularly to the simulation of air flow in a time-dependent domain representing vocal tract and vocal folds vibrations. We compare results obtained with the aid of linear and nonlinear elasticity models. The results show that it is more adequate to use the nonlinear elasticity St. Venant-Kirchhoff model in contrast to the linear elasticity model.

The subject of the paper is the brief description of the stability analysis of the space-time discontinuous Galerkin method for the numerical solution of a model nonlinear parabolic problem in a time dependent domain. The analyzed method is used for the solution of compressible flow in a time-dependent domain and interacted with an elastic body, applied to the simulation of vocal fold flow-induced vibrations.

The goal of this paper is the numerical simulation of vocal folds vibration excited by compressible viscous flow. It is necessary to describe the flow by the compressible Navier-Stokes equations in a time-dependent domain, solved by the ALE method. The vibrations of the vocal folds are modelled by the nonlinear dynamic St. Venant-Kirchhoff elasticity theory. For the flow problem we employ the discretization by the space-time discontinuous Galerkin method (STDGM) and for the elasticity problem the backward difference formula in time and discontinuous Galerkin method in space (BDF-DGM). The flow and elasticity problems are strongly coupled. We show that it is more appropriate to use the nonlinear elasticity model in contrast to linear elasticity model.

This article is concerned with the analysis of the discontinuous Galerkin method (DGM) for the numerical solution of an elliptic boundary value problem with a nonlinear Newton boundary condition in a two-dimensional polygonal domain. The growth of the nonlinearity is not compatible with the differential equation, which represents an obstacle in the analysis of the problem. Using monotone operator theory, it is possible to prove the existence and uniqueness of the weak solution and the approximate DG solution. The main emphasis is on the study of error estimates. To this end, the regularity of the weak solution is investigated, and it is shown that due to the singular boundary points, the solution loses regularity in the vicinity of these points. It transpires that the error estimation depends essentially on the opening angle of the corner points and the nonlinearity in the boundary term. It also depends on the parameter defining the nonlinear behaviour of the Newton boundary condition. At the end of this article, some computational experiments are presented.

This paper is concerned with the analysis of the finite element method for the numerical solution of an elliptic boundary value problem with a nonlinear Newton boundary condition in a two-dimensional polygonal domain. The weak solution loses regularity in a neighbourhood of boundary singularities, which may be at corners or at roots of the weak solution on edges. The main attention is paid to the study of error estimates. It turns out that the order of convergence is not dampened by the nonlinearity if the weak solution is nonzero on a large part of the boundary. If the weak solution is zero on the whole boundary, the nonlinearity only slows down the convergence of the function values but not the convergence of the gradient. The same analysis is carried out for approximate solutions obtained by numerical integration. The theoretical results are verified by numerical experiments.

The subject of the paper is the numerical solution of dynamic elasticity problems. We consider linear model and nonlinear Neo-Hookean model. First the continuous dynamic elasticity problem is formulated and then we pay attention to the derivation of the discrete problem. The space discretization is carried out by the discontinuous Galerkin method (DGM). It is combined with the backward difference formula (BDF) for the time discretization. Further, several numerical experiments are presented showing the behaviour of the developed numerical method in dependence on the coefficient in the penalty form. At the end the developed method is applied to the simulation of vibrations of 2D model of human vocal fold formed by four layers with different materials.

The subject of the paper is the numerical simulation of two-phase flow of immiscible fluids. Their motion is described by the incompressible Navier-Stokes equations with different constant density and viscosity for different fluids. The interface between the fluids is defined with the aid of the level-set method using a transport first-order hyperbolic equation. The Navier-Stokes system is equipped with initial and boundary conditions and transmission conditions on the interface between the two fluids. This system is discretized by the Taylor-Hood P2/P1 conforming finite elements in space and the second-order BDF method in time. The transport level-set problem is solved with the aid of the space-time discontinuous Galerkin method. Numerical experiments demonstrate that the developed method is accurate and robust.

The paper is concerned with the application of the space-time discontinuous Galerkin method (STDGM) to the numerical solution of the interaction of a compressible flow and an elastic structure. The flow is described by the system of compressible Navier-Stokes equations written in the conservative form. They are coupled with the dynamic elasticity system of equations describing the deformation of the elastic body, induced by the aerodynamical force on the interface between the gas and the elastic structure. The domain occupied by the fluid depends on time. It is taken into account in the Navier-Stokes equations rewritten with the aid of the arbitrary Lagrangian-Eulerian (ALE) method. The resulting coupled system is discretized by the STDGM using piecewise polynomial approximations of the sought solution both in space and time. The developed method can be applied to the solution of the compressible flow for a wide range of Mach numbers and Reynolds numbers. For the simulation of elastic deformations two models are used: the linear elasticity model and the nonlinear neo-Hookean model. The main goal is to show the robustness and applicability of the method to the simulation of the air flow in a simplified model of human vocal tract and the flow induced vocal folds vibrations. It will also be shown that in this case the linear elasticity model is not adequate and it is necessary to apply the nonlinear model.

The novelty of the paper is the application of the discontinuous Galerkin method (DGM) to the solution of viscous compressible flow with low Mach number (M approximate to 10(-2) - 10(-3)). The compressible Navier-Stokes equations are written in the conservative form using conservative variables. They are discretized by the DGM in space. In time we use either the backward difference formula (BDF) or the discontinuous Galerkin method. The resulting technique is called the space-time discontinuous Galerkin method (STDGM). The goal of the presented analysis is to prove the ability of the developed DGM to solve viscous compressible low Mach number flows described by the compressible Navier-Stokes equations without any modification and to obtain results comparable with simulations of incompressible viscous flow described by the incompressible Navier-Stokes equations, solved by the finite element method (FEM). We present numerical results for a flow in a simplified channel of the human vocal tract, for a flow around the airfoil NACA0012 and moreover for flow-induced airfoil vibrations. The results prove the robustness of the discontinuous Galerkin method with respect to the low Mach number. (C) 2018 Elsevier Ltd. All rights reserved.

We assume the heat equation in a time dependent domain, where the evolution of the domain is described by a given mapping. The problem is discretized by the discontinuous Galerkin (DG) method in space as well as in time with the aid of Arbitrary Lagrangian-Eulerian (ALE) method. The sketch of the proof of the stability of the method is shown.

The paper is concerned with the numerical analysis of an elliptic equation in a polygon with a nonlinear Newton boundary condition, discretized by the finite element or discontinuous Galerkin methods. Using the monotone operator theory, it is possible to prove the existence and uniqueness of the exact weak solution and the approximate solution. The main attention is paid to the study of error estimates. To this end, the regularity of the weak solution is investigated and it is shown that due to the boundary corner points, the solution looses regularity in a vicinity of these points. It comes out that the error estimation depends essentially on the opening angle of the corner points and on the parameter defining the nonlinear behaviour of the Newton boundary condition. Theoretical results are compared with numerical experiments confirming a nonstandard behaviour of error estimates.

This paper is concerned with the numerical simulation of the interaction of compressible viscous flow with elastic structures. The flow is described by the compressible Navier-Stokes equations written in the arbitrary Lagrangian-Eulerian (ALE) form. For the elastic deformation we use 2D linear elasticity and nonlinear St. Venant-Kirchhoff and neo-Hookean models. The discretization of both flow problem and elasticity problem is realized by the discontinuous Galerkin finite element method (DGFEM). The main attention is paid to testing the DGFEM applied to the solution of elasticity problems. Then we present an example of the fluid-structure interaction (FSI). The applicability of the developed technique is demonstrated by several numerical experiments.

In this paper we investigate the stability of the space-time discontinuous Galerkin method (STDGM) for the solution of nonstationary, linear convection-diffusion-reaction problem in time-dependent domains formulated with the aid of the arbitrary Lagrangian-Eulerian (ALE) method. At first we define the continuous problem and reformulate it using the ALE method, which replaces the classical partial time derivative with the so called ALE-derivative and an additional convective term. In the second part of the paper we discretize our problem using the space-time discontinuous Galerkin method. The space discretization uses piecewise polynomial approximations of degree p ≥ 1, in time we use only piecewise linear discretization. Finally in the third part of the paper we present our results concerning the unconditional stability of the method.

This paper is concerned with the numerical solution of dynamic elasticity by the discontinuous Galerkin (dG) method. We consider the linear and nonlinear St. Venant-Kirchhoff model. The dynamic elasticity problem is split into two systems of first order in time. They are discretized by the discontinuous Galerkin method in space and backward difference formula in time. The developed method is tested by numerical experiments. Then the method is combined with the space-time dG method for the solution of compressible flow in a time dependent domain and used for the numerical simulation of fluid-structure interaction.

In this paper we investigate the stability of the space-time discontinuous Galerkin method (STDGM) for the solution of nonstationary, linear convection-diffusion-reaction problem in time-dependent domains formulated with the aid of the arbitrary Lagrangian-Eulerian (ALE) method. The stability is uniform with respect to the diffusion coefficient. The ALE method replaces the classical partial time derivative with the so called ALE-derivative and an additional convective term. In the second part of the paper we discretize our problem using the space-time discontinuous Galerkin method. In the formulation of the numerical scheme we use the nonsymmetric, symmetric and incomplete versions of the space discretization of diffusion terms and interior and boundary penalty. The space discretization uses piecewise polynomial approximations of degree p >= 1, in time we use only piecewise linear discretization. Finally in the third part of the paper we present our results concerning the uniform unconditional stability of the method.

The subject of the paper is the numerical analysis of the transport level-set problem discretized by the discontinuous Galerkin method. Without the assumption that the first order nonstationary transport equation contains a reaction term, which is used in a standard literature, we prove error estimates in the L∞(L2)-norm in the case of the space semidiscretization method of lines and in the case of the space–time discontinuous Galerkin method in the L2(L2)-norm. Numerical experiments support the derived error estimates and show that they are not sharp in the case of the space–time discontinuous Galerkin method.

The subject of this paper is the analysis of the space-time discontinuous Galerkin method for the solution of nonstationary, nonlinear, convection-diffusion problems. In the formulation of the numerical scheme, the nonsymmetric, symmetric and incomplete versions of the discretization of diffusion terms and interior and boundary penalty are used. Then error estimates are briefly characterized. The main attention is paid to the investigation of unconditional stability of the method. An important tool is the concept of the discrete characteristic function. Theoretical results are accompanied by numerical experiments.

In this chapter we introduce two types of the DG discretization that were derived with the aid of a mixed formulation.