Two-Grid Method for Burgers’ Equation by a New Mixed Finite Element Scheme

In this article, we present two-grid stable mixed finite element method for the 2D Burgers’ equation approximated by the -P1 pair which satisfies the inf–sup condition. This method consists in dealing with the nonlinear system on a coarse mesh with width H and the linear system on a fine mesh with width h << H by using Crank–Nicolson time-discretization scheme. Our results show that if we choose H2 = h this method can achieve asymptotically optimal approximation. Error estimates are derived in detail. Finally, numerical experiments show the efficiency of our proposed method and justify the theoretical results

Geometric Numerical Integration (hybrid meeting)

The topics of the workshop included interactions between geometric numerical integration and numerical partial differential equations; geometric aspects of stochastic differential equations; interaction with optimisation and machine learning; new applications of geometric integration in physics; problems of discrete geometry, integrability, and algebraic aspects

Space-time Methods for Time-dependent Partial Differential Equations

Modern discretizations of time-dependent PDEs consider the full problem in the space-time cylinder and aim to overcome limitations of classical approaches such as the method of lines (first discretize in space and then solve the resulting ODE) and the Rothe method (first discretize in time and then solve the PDE). A main advantage of a holistic space-time method is the direct access to space-time adaptivity and to the backward problem (required for the dual problem in optimization or error control). Moreover, this allows for parallel solution strategies simultaneously in time and space. Several space-time concepts where proposed (different conforming and nonconforming space-time finite elements, the parareal method, wavefront relaxation etc.) but this topic has become a rapidly growing field in numerical analysis and scientific computing. In this workshop the focus is the development of adaptive and flexible space-time discretization methods for solving parabolic and hyperbolic space-time partial differential equations

Numerical Methods for Partial Differential Equations

These lecture notes are devoted to the numerical solution of partial differential equations (PDEs). PDEs arise in many fields and are extremely important in modeling of technical processes with applications in physics, biology, chemisty, economics, mechanical engineering, and so forth. In these notes, not only classical topics for linear PDEs such as finite differences, finite elements, error estimation, and numerical solution schemes are addressed, but also schemes for nonlinear PDEs and coupled problems up to current state-of-the-art techniques are covered. In the Winter 2020/2021 an International Class with additional funding from DAAD (German Academic Exchange Service) and local funding from the Leibniz University Hannover, has led to additional online materials such as links to youtube videos, which complement these lecture notes. This is the updated and extended Version 2. The first version was published under the DOI: https://doi.org/10.15488/9248

Final report on the Copper Mountain conference on multigrid methods

The Copper Mountain Conference on Multigrid Methods was held on April 6-11, 1997. It took the same format used in the previous Copper Mountain Conferences on Multigrid Method conferences. Over 87 mathematicians from all over the world attended the meeting. 56 half-hour talks on current research topics were presented. Talks with similar content were organized into sessions. Session topics included: fluids; domain decomposition; iterative methods; basics; adaptive methods; non-linear filtering; CFD; applications; transport; algebraic solvers; supercomputing; and student paper winners

Modeling studies and numerical analyses of coupled PDEs system in electrohydrodynamics

Electrohydrodynamics (EHD) is the term used for the hydrodynamics coupled with electrostatics, whose governing equations consist of the electrostatic potential (Poisson) equation, the ionic concentration (Nernst-Planck) equations, and Navier-Stokes equations for an incompressible, viscous dielectric liquid. In this dissertation, we focus on a specic application of EHD - fuel cell dynamics - in the eld of renewable and clean energy, study its traditional model and attempt to develop a new fuel cell model based on the traditional EHD model. Meanwhile, we develop a series of ecient and robust numerical methods for these models, and carry out their numerical analyses on the approximation accuracy. In particular, we analyze the error estimates of nite element method for a simplied 2D isothermal steady state two-phase transport model of Proton Exchange Membrane Fuel Cell (PEMFC) as well as its transient version. On the aspect of hydrodynamics arising in the fuel cell system, the fluid flow through the open channels and porous media at the same time, both Navier-Stokes equations and Darcy\u27s law are involved in the fluid domains, leading to a Navier-Stokes-Darcy coupling problem. In this dissertation, we study a one-continuum model approach, so-called Brinkman model, to overcome this problem in a more ecient way. To develop a new fuel cell model based on EHD theory, in addition to the two-phase transport model of fuel cells, we carry out numerical analyses for Poisson-Nernst-Planck (PNP) equations using both standard FEM and mixed FEM, which are the essential governing equations involved by EHD model. Finally, we are able to further extend the traditional fuel cell model to more general cases in view of EHD characteristics, and develop a new fuel cell model by appropriately combining PNP equations with the traditional fuel cell model. We conduct the error analysis for PNP-Brinkman system in this dissertation

Hybridizable compatible finite element discretizations for numerical weather prediction: implementation and analysis

There is a current explosion of interest in new numerical methods for atmospheric modeling. A driving force behind this is the need to be able to simulate, with high efficiency, large-scale geophysical flows on increasingly more parallel computer systems. Many current operational models, including that of the UK Met Office, depend on orthogonal meshes, such as the latitude-longitude grid. This facilitates the development of finite difference discretizations with favorable numerical properties. However, such methods suffer from the ``pole problem," which prohibits the model to make efficient use of a large number of computing processors due to excessive concentration of grid-points at the poles. Recently developed finite element discretizations, known as ``compatible" finite elements, avoid this issue while maintaining the key numerical properties essential for accurate geophysical simulations. Moreover, these properties can be obtained on arbitrary, non-orthogonal meshes. However, the efficient solution of the resulting discrete systems depend on transforming the mixed velocity-pressure (or velocity-pressure-buoyancy) system into an elliptic problem for the pressure. This is not so straightforward within the compatible finite element framework due to inter-element coupling. This thesis supports the proposition that systems arising from compatible finite element discretizations can be solved efficiently using a technique known as ``hybridization." Hybridization removes inter-element coupling while maintaining the desired numerical properties. This permits the construction of sparse, elliptic problems, for which fast solver algorithms are known, using localized algebra. We first introduce the technique for compatible finite element discretizations of simplified atmospheric models. We then develop a general software abstraction for the rapid implementation and composition of hybridization methods, with an emphasis on preconditioning. Finally, we extend the technique for a new compatible method for the full, compressible atmospheric equations used in operational models.Open Acces

A Coupled Stochastic-Deterministic Method for the Numerical Solution of Population Balance Systems

In this thesis, a new algorithm for the numerical solution of population balance systems is proposed and applied within two simulation projects. The regarded systems stem from chemical engineering. In particular, crystallization processes in fluid environment are regarded. The descriptive population balance equations are extensions of the classical Smoluchowski coagulation equation, of which they inherit the numerical difficulties introduced with the coagulation integral, especially in regard of higher dimensional particle models. The new algorithm brings together two different fields of numerical mathematics and scientific computing, namely a stochastic particle simulation based on a Markov process Monte—Carlo method, and (deterministic) finite element schemes from computational fluid dynamics. Stochastic particle simulations are approved methods for the solution of population balance equations. Their major advantages are the inclusion of microscopic information into the model while offering convergence against solutions of the macroscopic equation, as well as numerical efficiency and robustness. The embedding of a stochastic method into a deterministic flow simulation offers new possibilities for the solution of coupled population balance systems, especially in regard of the microscopic details of the interaction of particles. In the thesis, the new simulation method is first applied to a population balance system that models an experimental tube crystallizer which is used for the production of crystalline aspirin. The device is modeled in an axisymmetric two-dimensional fashion. Experimental data is reproduced in moderate computing time. Thereafter, the method is extended to three spatial dimensions and used for the simulation of an experimental, continuously operated fluidized bed crystallizer. This system is fully instationary, the turbulent flow is computed on-the-fly. All the used methods from the simulation of the Navier—Stokes equations, the simulation of convection-diffusion equations, and of stochastic particle simulation are introduced, motivated and discussed extensively. Coupling phenomena in the regarded population balance systems and the coupling algorithm itself are discussed in great detail. Furthermore, own results about the efficient numerical solution of the Navier—Stokes equations are presented, namely an assessment of fast solvers for discrete saddle point problems, and an own interpretation of the classical domain decompositioning method for the parallelization of the finite element method

A Fully Parallelized and Budgeted Multi-level Monte Carlo Framework for Partial Differential Equations: From Mathematical Theory to Automated Large-Scale Computations

All collected data on any physical, technical or economical process is subject to uncertainty. By incorporating this uncertainty in the model and propagating it through the system, this data error can be controlled. This makes the predictions of the system more trustworthy and reliable. The multi-level Monte Carlo (MLMC) method has proven to be an effective uncertainty quantification tool, requiring little knowledge about the problem while being highly performant. In this doctoral thesis we analyse, implement, develop and apply the MLMC method to partial differential equations (PDEs) subject to high-dimensional random input data. We set up a unified framework based on the software M++ to approximate solutions to elliptic and hyperbolic PDEs with a large selection of finite element methods. We combine this setup with a new variant of the MLMC method. In particular, we propose a budgeted MLMC (BMLMC) method which is capable to optimally invest reserved computing resources in order to minimize the model error while exhausting a given computational budget. This is achieved by developing a new parallelism based on a single distributed data structure, employing ideas of the continuation MLMC method and utilizing dynamic programming techniques. The final method is theoretically motivated, analyzed, and numerically well-tested in an automated benchmarking workflow for highly challenging problems like the approximation of wave equations in randomized media

On the convergence order of the finite element error in the kinetic energy for high Reynolds number incompressible flows

The kinetic energy of a flow is proportional to the square of the norm of the velocity. Given a sufficient regular velocity field and a velocity finite element space with polynomials of degree , then the best approximation error in is of order . In this survey, the available finite element error analysis for the velocity error in is reviewed, where is a final time. Since in practice the case of small viscosity coefficients or dominant convection is of particular interest, which may result in turbulent flows, robust error estimates are considered, i.e., estimates where the constant in the error bound does not depend on inverse powers of the viscosity coefficient. Methods for which robust estimates can be derived enable stable flow simulations for small viscosity coefficients on comparatively coarse grids, which is often the situation encountered in practice. To introduce stabilization techniques for the convection-dominated regime and tools used in the error analysis, evolutionary linear convection–diffusion equations are studied at the beginning. The main part of this survey considers robust finite element methods for the incompressible Navier–Stokes equations of order , , and for the velocity error in . All these methods are discussed in detail. In particular, a sketch of the proof for the error bound is given that explains the estimate of important terms which determine finally the order of convergence. Among them, there are methods for inf–sup stable pairs of finite element spaces as well as for pressure-stabilized discretizations. Numerical studies support the analytic results for several of these methods. In addition, methods are surveyed that behave in a robust way but for which only a non-robust error analysis is available. The conclusion of this survey is that the problem of whether or not there is a robust method with optimal convergence order for the kinetic energy is still open