Space-time discontinuous Galerkin method for wet-chemical etching of microstructures

In this paper we discuss the application of a space-time discontinuous Galerkin finite element method for convection-diffusion problems to the simulation of wet-chemical etching of microstructures. In the space-time DG method no distinction is made in the discretization between the space and time variables and discontinuous basis functions are used both in space and time. This approach results in an efficient numerical technique to deal with time-dependent flow domains as occur in wet-chemical etching, while maintaining a fully conservative discretization. The method offers great flexibility in mesh adaptation and special attention is given to the generation of an initial solution and mesh when there is no etching cavity yet. Numerical simulations of the etching of a two-dimensional slit are discussed for different regimes, namely diffusion-controlled and convection-dominated etching. These results show good agreement with analytical results in the diffusion-controlled regime. Using a simple model for the fluid velocity the typical asymmetric etching cavities are obtained in the convection dominated regime and the results agree qualitatively well with those obtained from full Navier-Stokes simulations

Multigrid optimization for space-time discontinuous Galerkin discretizations of advection dominated flows

The goal of this research is to optimize multigrid methods for higher order accurate space-time discontinuous Galerkin discretizations. The main analysis tool is discrete Fourier analysis of two- and three-level multigrid algorithms. This gives the spectral radius of the error transformation operator which predicts the asymptotic rate of convergence of the multigrid algorithm. In the optimization process we therefore choose to minimize the spectral radius of the error transformation operator. We specifically consider optimizing h-multigrid methods with explicit Runge-Kutta type smoothers for second and third order accurate space-time discontinuous Galerkin finite element discretizations of the 2D advection-diffusion equation. The optimized schemes are compared with current h-multigrid techniques employing Runge-Kutta type smoothers. Also, the efficiency of h-, p- and hp-multigrid methods for solving the Euler equations of gas dynamics with a higher order accurate space-time DG method is investigated

A space-time discontinuous Galerkin method for the time-dependent Oseen equations

A space–time discontinuous Galerkin finite element method for the Oseen equations on time-dependent flow domains is presented. The algorithm results in a higher order accurate conservative discretization on moving and deforming meshes and is well suited for hp-adaptation. A detailed analysis of the stability of the numerical discretization is given which shows that the algorithm is unconditionally stable, also when equal order polynomial basis functions for the pressure and velocity are used. The accuracy of the space–time discretization is investigated using a detailed hp-error analysis and computations on a model problem.\ud \u

Well Posed Problems and Boundary Conditions in Computational Fluid Dynamics

All numerical calculations will fail to provide a reliable answer unless the continuous problem under consideration is well posed. Well-posedness depends in most cases only on the choice of boundary conditions. In this paper we will highlight this fact by discussing well-posedness of the most important equations in computational uid dynamics, namely the time-dependent compressible Navier-Stokes equations.   In particular, we will discuss i) how many boundary conditions are required, ii) where to impose them and iii) which form they should have. The procedure is based on the energy method and generalizes the characteristic boundary procedure for the Euler equations to the compressible Navier-Stokes equations.   Once the boundary conditions in terms of i-iii) are known, one issue remains; they can be imposed weakly or strongly. The weak and strong imposition is discussed for the continuous case. It will be shown that the weak and strong boundary procedures produce identical solutions and that the boundary conditions are satised exactly also in the weak procedure.   We conclude by relating the well-posedness results to energy-stability of the numerical approximation. It is shown that the results obtained in the well-posedness analysis for the continuous problem generalizes directly to stability of the discrete problem

A study on discontinuous Galerkin finite element methods for elliptic problems

In this report we study several approaches of the discontinuous Galerkin finite element methods for elliptic problems. An important aspect in these formulations is the use of a lifting operator, for which we present an efficient numerical approximation technique. Numerical experiments for two different discontinuous Galerkin methods are presented for one dimensional problems and compared with exact results. In addition, the theoretical order of accuracy is verified numerically

Space-time discontinuous Galerkin method for wet-chemical etching of microstructures

In this paper we discuss the application of a space-time discontinuous Galerkin finite element method for convection-diffusion problems to the simulation of wet-chemical etching of microstructures. In the space-time DG method no distinction is made in the discretization between the space and time variables and discontinuous basis functions are used both in space and time. This approach results in an efficient numerical technique to deal with time-dependent flow domains as occur in wet-chemical etching, while maintaining a fully conservative discretization. The method offers great flexibility in mesh adaptation and special attention is given to the generation of an initial solution and mesh when there is no etching cavity yet. Numerical simulations of the etching of a two-dimensional slit are discussed for different regimes, namely diffusion-controlled and convection-dominated etching. These results show good agreement with analytical results in the diffusion-controlled regime. Using a simple model for the fluid velocity the typical asymmetric etching cavities are obtained in the convection dominated regime and the results agree qualitatively well with those obtained from full Navier-Stokes simulations