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- ItemStability of explicit Runge-Kutta methods for high order finite element approximation of linear parabolic equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Huang, Weizhang; Kamenski, Lennard; Lang, JensWe study the stability of explicit Runge-Kutta methods for high order Lagrangian finite element approximation of linear parabolic equations and establish bounds on the largest eigenvalue of the system matrix which determines the largest permissible time step. A bound expressed in terms of the ratio of the diagonal entries of the stiffness and mass matrices is shown to be tight within a small factor which depends only on the dimension and the choice of the reference element and basis functions but is independent of the mesh or the coefficients of the initial-boundary value problem under consideration. Another bound, which is less tight and expressed in terms of mesh geometry, depends only on the number of mesh elements and the alignment of the mesh with the diffusion matrix. The results provide an insight into how the interplay between the mesh geometry and the diffusion matrix affects the stability of explicit integration schemes when applied to a high order finite element approximation of linear parabolic equations on general nonuniform meshes.
- ItemError estimates in weighted Sobolev norms for finite element immersed interface methods(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Heltai, Luca; Rotundo, NellaWhen solving elliptic partial differential equations in a region containing immersed interfaces (possibly evolving in time), it is often desirable to approximate the problem using a uniform background discretisation, not aligned with the interface itself. Optimal convergence rates are possible if the discretisation scheme is enriched by allowing the discrete solution to have jumps aligned with the surface, at the cost of a higher complexity in the implementation. A much simpler way to reformulate immersed interface problems consists in replacing the interface by a singular force field that produces the desired interface conditions, as done in immersed boundary methods. These methods are known to have inferior convergence properties, depending on the global regularity of the solution across the interface, when compared to enriched methods. In this work we prove that this detrimental effect on the convergence properties of the approximate solution is only a local phenomenon, restricted to a small neighbourhood of the interface. In particular we show that optimal approximations can be constructed in a natural and inexpensive way, simply by reformulating the problem in a distributionally consistent way, and by resorting to weighted norms when computing the global error of the approximation.
- ItemModelling and simulation of flame cutting for steel plates with solid phases and melting(Berlin ; Heidelberg : Springer, 2020) Arenas, Manuel J.; Hömberg, Dietmar; Lasarzik, Robert; Mikkonen, Pertti; Petzold, ThomasThe goal of this work is to describe in detail a quasi-stationary state model which can be used to deeply understand the distribution of the heat in a steel plate and the changes in the solid phases of the steel and into liquid phase during the flame cutting process. We use a 3D-model similar to previous works from Thiébaud (J. Mater. Process. Technol. 214(2):304–310, 2014) and expand it to consider phases changes, in particular, austenite formation and melting of material. Experimental data is used to validate the model and study its capabilities. Parameters defining the shape of the volumetric heat source and the power density are calibrated to achieve good agreement with temperature measurements. Similarities and differences with other models from literature are discussed. © 2020, The Author(s).
- ItemSome analytical results for an algebraic flux correction scheme for a steady convection-diffusion equation in 1D(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Barrenechea, Gabriel R.; John, Volker; Knobloch, PetrAlgebraic flux correction schemes are nonlinear discretizations of convection dominated problems. In this work, a scheme from this class is studied for a steady-state convection-diffusion equation in one dimension. It is proved that this scheme satisfies the discrete maximum principle. Also, as it is a nonlinear scheme, the solvability of the linear subproblems arising in a Picard iteration is studied, where positive and negative results are proved. Furthermore, the non-existence of solutions for the nonlinear scheme is proved by means of counterexamples. Therefore, a modification of the method, which ensures the existence of a solution, is proposed. A weak version of the discrete maximum principle is proved for this modified method.
- ItemEigensolutions of the Wigner-Eisenbud problem for a cylindrical nanowire within finite volume method(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Racec, Paul N.; Schade, Stanley; Kaiser, Hans-ChristophWe present a finite volume method for computing a representative range of eigenvalues and eigenvectors of the Schrödinger operator on a three dimensional cylindrically symmetric bounded domain with mixed boundary conditions. More specifically, we deal with a semiconductor nanowire which consists of a dominant host material and contains heterostructure features such as double-barriers or quantum dots. The three dimensional Schrödinger operator is reduced to a family of two dimensional Schrödinger operators distinguished by a centrifugal potential. Ultimately, we numerically treat them by means of a finite volume method. We consider a uniform, boundary conforming Delaunay mesh, which additionally conforms to the material interfaces. The 1/r singularity is eliminated by approximating r at the vertexes of the Voronoi boxes. We study how the anisotropy of the effective mass tensor acts on the uniform approximation of the first K eigenvalues and eigenvectors and their sequential arrangement. There exists an optimal uniform Delaunay discretization with matching anisotropy. This anisotropic discretization yields best accuracy also in the presence of a mildly varying scattering potential, shown exemplarily for a nanowire resonant tunneling diode. For potentials with 1/r singularity one retrieves the theoretically established first order convergence, while the second order convergence is recovered only on uniform grids with an anisotropy correction.
- ItemStability of explicit Runge-Kutta methods for finite element approximation of linear parabolic equations on anisotropic meshes(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Huang, Weizhang; Kamenski, Lennard; Lang, JensWe study the stability of explicit Runge-Kutta integration schemes for the linear finite element approximation of linear parabolic equations. The derived bound on the largest permissible time step is tight for any mesh and any diffusion matrix within a factor of 2(d + 1), where d is the spatial dimension. Both full mass matrix and mass lumping are considered. The bound reveals that the stability condition is affected by two factors. The first one depends on the number of mesh elements and corresponds to the classic bound for the Laplace operator on a uniform mesh. The other factor reflects the effects of the interplay of the mesh geometry and the diffusion matrix. It is shown that it is not the mesh geometry itself but the mesh geometry in relation to the diffusion matrix that is crucial to the stability of explicit methods. When the mesh is uniform in the metric specified by the inverse of the diffusion matrix, the stability condition is comparable to the situation with the Laplace operator on a uniform mesh. Numerical results are presented to verify the theoretical findings.
- ItemA posteriori error control for stationary coupled bulk-surface equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Eigel, Martin; Müller, RüdigerWe consider a system of two coupled elliptic equations, one defined on a bulk domain and the other one on the boundary surface. Problems of this kind of problem are relevant for applications in engineering, chemistry and in biology like e.g. biological signal transduction. For the a posteriori error control of the coupled system, a residual error estimator is derived which takes into account the approximation errors due to the finite element discretisation in space as well as the polyhedral approximation of the surface. An adaptive refinement algorithm controls the overall error. Numerical experiments illustrate the performance of the a posteriori error estimator and the proposed adaptive algorithm with several benchmark examples.
- ItemOptimal control of buoyancy-driven liquid steel stirring modeled with single-phase Navier–Stokes equations(Berlin ; Heidelberg : Springer, 2021) Wilbrandt, Ulrich; Alia, Najib; John, VolkerGas stirring is an important process used in secondary metallurgy. It allows to homogenize the temperature and the chemical composition of the liquid steel and to remove inclusions which can be detrimental for the end-product quality. In this process, argon gas is injected from two nozzles at the bottom of the vessel and rises by buoyancy through the liquid steel thereby causing stirring, i.e., a mixing of the bath. The gas flow rates and the positions of the nozzles are two important control parameters in practice. A continuous optimization approach is pursued to find optimal values for these control variables. The effect of the gas appears as a volume force in the single-phase incompressible Navier–Stokes equations. Turbulence is modeled with the Smagorinsky Large Eddy Simulation (LES) model. An objective functional based on the vorticity is used to describe the mixing in the liquid bath. Optimized configurations are compared with a default one whose design is based on a setup from industrial practice.
- ItemA local projection stabilization finite element method with nonlinear crosswind diffusion for convection-diffusion-reaction equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Barrenechea, Gabriel R.; John, Volker; Knobloch, PetrAn extension of the local projection stabilization (LPS) finite element method for convection-diffusion-reaction equations is presented and analyzed, both in the steady-state and the transient setting. In addition to the standard LPS method, a nonlinear crosswind diffusion term is introduced that accounts for the reduction of spurious oscillations. The existence of a solution can be proved and, depending on the choice of the stabilization parameter, also its uniqueness. Error estimates are derived which are supported by numerical studies. These studies demonstrate also the reduction of the spurious oscillations.