Doktorandenseminar des WIAS
FG3: Numerische Mathematik und Wissenschaftliches
RG3: Numerical Mathematics
and Scientific Computing/
LG5: Numerik für innovative
LG5: Numerics for
innovative semiconductor devices
Seminar Numerische Mathematik / Numerical mathematics seminars
aktuelles Programm / current program
Hybrid-/Online-Vorträge finden über ''Zoom'' statt. Der Zoom-Link wird jeweils ca. 15 Minuten vor Beginn des Gesprächs versendet. / The zoom link will be send about 15 minutes before the start of the talk. People who are not members of the research group 3 and who are interested in participating should contact firstname.lastname@example.org to obtain the zoom login details.
Donnerstag, 16. 05. 2023, 13:30 Uhr (WIAS-ESH)
Camilla Belponer (WIAS & Uni Augsburg)
Reduced Lagrange multiplier approach for the non-matching coupled problems in multiscale elasticity
The interest in efficient simulation of vascularized tissues is motivated by the solution of inverse problems in the context of tissue imaging, where available medical data (such as those obtained via Magnetic Resonance Elastography) have a limited resolution, typically at the scale of an effective - macro scale - tissue, and cannot resolve the microscale of quantities of interests related, for instance, to the tissue vasculature. Our model is based on a geometrical multiscale 3D (elastic) -1D (fluid) formulation combined with an immersed method. At the elastic-fluid immersed boundary \(\Gamma\) we impose a trace-averaged boundary condition whose goal is to impose only a local Dirichlet boundary condition on the tissue-vessel interface allowing the enforcement of a pure normal displacement at the fictional vessel boundary. In order to decouple the discretization of the elastic tissue from the vessel boundary, the boundary condition on \(\Gamma\) is imposed via a Lagrange multiplier, modeling the fluid vessels as immersed singular sources for the elasticity equation. Next, to efficiently handle the multiscale nature of the problem, the problem is formulated as a mixed-dimensional PDE using the recently proposed framework of reduced Lagrange multipliers on a space of co-dimension 2. In this talk, we present the numerical analysis of the obtained formulation and we discuss accuracy properties and convergences of the method, validating it in several numerical examples. Finally, we present perspectives for the coupling with a one-dimensional flow model defined on the vascular network and for the numerical upscaling of the tissue model.
Donnerstag, 11. 05. 2023, 14:00 Uhr (WIAS-ESH/Hybrid Event)
Théo Belin (Centrale Supélec)
An entropic finite volumes scheme for a fully non-linear forward-backward parabolic equation
We investigate a fully non-linear forward-backward parabolic equation, stemming from a thermodynamical model of phase change in nylon. The non-monotonicity of the non-linearity yields an ill-posed problem in any standard framework for parabolic equations. To overcome this difficulty one can introduce a singular perturbation of the problem called the Sobolev approximation. In the singular limit, entropy solutions are naturally unveiled and have first been studied by Plotnikov and then later in restricted settings by Evans and Porthilheiro, Corrado and others. These entropy solutions exhibit a free discontinuity and a hysteretic behaviour which we describe. While existence of such solutions is guaranteed by time-discrete approximations, the uniqueness of these entropy solutions has yet to be precisely studied in higher dimensions. Because of the ill-posed nature of the problem, the convergence of numerical approximations of the original equation is not yet attained. It is an open problem to decide wether any of these direct schemes can select an entropy solution in the limit. This is why we present an entropy-preserving finite volume scheme which tracks the hysteresis behaviour through a phase indicator function. The estimates proven so far do not directly allow for the identification of the limit as an entropy solution. A missing link to the convergence is the \(L^1\) compactness either for the solution or for the phase indicator function. Numerical simulations are shown.
Donnerstag, 23. 03. 2023, 14:00 Uhr (WIAS-ESH/Hybrid Event)
Gabriel Barrenechea (University of Strathclyde)
Positivity-preserving discretisations in general meshes
The quest for physical consistency in the discretisation of PDEs started as soon as the numerical methods started being proposed. By physical consistency we mean a discretisation that by design satisfies a property also satisfied by the continuous PDE. This property might be positivity of the discrete solution, or preservation of some bounds (e.g., concentrations should belong to the interval [0, 1]), or can also be energy preservation, or exactly divergence-free velocities for incompressible fluids. Regarding positivity preservation, this topic has been around since the pioneering work by Ph. Ciarlet in the late 1960s and early 1970s. In the context of finite element methods, it was shown in those early works (and not significantly improved since), that in order for a finite element method to preserve positivity the mesh needs to satisfy certain geometrical restrictions, e.g., in two space dimensions with simplicial elements the triangulation needs to be of Delaunay type (in higher dimensions or quadrilateral meshes the restrictions are more involved). Throughout the years several conclusions have been reached in this topic, but in the context of finite element methods the discretisations tend to be of first order in space. So, many important problems still remain open. In particular, one open problem is how to build a discretisation that will lead to a positive solution regarless of the geometry of the mesh and the order of the finite element method. In this talk I will review recent results addressing the last question posed in the last paragraph. More precisely, I will present a method that enforces bound-preservation (at the degrees of freedom) of the discrete solution. The method is built by first defining an algebraic projection onto the convex closed set of finite element functions that satisfy the bounds given by the solution of the PDE. Then, this projection is hardwired into the definition of the method by writing a discrete problem posed for this projected part of the solution. Since this process is done independently of the shape of the basis functions, and no result on the resulting finite element matrix is used, then the outcome is a finite element function that satisfies the bounds at the degrees of freedom. Another important observation to make is that this approach is related to variational inequalities, and this fact will be exploited in the error analysis. The core of the talk will be devoted to explaining the main idea in the context of linear (and nonlinear) reaction-diffusion equations. Then, I will explain the main difficulties encountered when extending this method to convection-diffusion equations, and, more importantly, to a finite element method defined in polytopal meshes. The results in this talk have been carried out in collaboration with Abdolreza Amiri (Strathclyde, UK), Emmanuil Geourgoulis (Heriot-Watt, UK and Athens, Greece), Tristan Pryer (Bath, UK), and Andreas Veeser (Milan, Italy).
Donnerstag, 19. 01. 2023, 14:00 Uhr (WIAS-ESH/Hybrid Event)
Michele Pütz (BTU Cottbus-Senftenberg)
Quadrature-Based Moment Methods for the Solution of Population Balance Equations
Particulate systems of many kinds are mathematically described by a multivariate number density function (NDF) whose evolution is governed by a so-called population balance equation (PBE). Typically, such equations are characterized by a high dimensionality and, accordingly, high computational costs for the numerical solution. Quadrature-based moment methods (QBMMs) aim to efficiently approximate solutions to PBEs by solving only for a set of moments instead of the NDF and closing the PBE using Gaussian quadrature rules. The upcoming presentation will give an introduction to QBMMs, elaborate on limitations and challenges associated with non-smooth terms in the moment equations using physical examples and provide some results related to the performance and accuracy of QBMMs.
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