Schnelle Löser für Partielle Differentialgleichungen

The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch (Leipzig), and Gabriel Wittum (Frankfurt am Main), was held May 22nd–May 28th, 2011. This meeting was well attended by 54 participants with broad geographic representation from 7 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds

Function spaces and functional frameworks

The goal is to provide an overview about function spaces, and more generally speaking functional frameworks that include metric spacs, normed spaces, inner product spaces, and convex sets for variational inequalities. Throughout, the implication to algorithms and practical applications is made and sometimes illustrated with numerical simulations from my own work

Multiscale Methods for Stochastic Collocation of Mixed Finite Elements for Flow in Porous Media

This thesis contains methods for uncertainty quantification of flow in porous media through stochastic modeling. New parallel algorithms are described for both deterministic and stochastic model problems, and are shown to be computationally more efficient than existing approaches in many cases.First, we present a method that combines a mixed finite element spatial discretization with collocation in stochastic dimensions on a tensor product grid. The governing equations are based on Darcy's Law with stochastic permeability. A known covariance function is used to approximate the log permeability as a truncated Karhunen-Loeve expansion. A priori error analysis is performed and numerically verified.Second, we present a new implementation of a multiscale mortar mixed finite element method. The original algorithm uses non-overlapping domain decomposition to reformulate a fine scale problem as a coarse scale mortar interface problem. This system is then solved in parallel with an iterative method, requiring the solution to local subdomain problems on every interface iteration. Our modified implementation instead forms a Multiscale Flux Basis consisting of mortar functions that represent individual flux responses for each mortar degree of freedom, on each subdomain independently. We show this approach yields the same solution as the original method, and compare the computational workload with a balancing preconditioner.Third, we extend and combine the previous works as follows. Multiple rock types are modeled as nonstationary media with a sum of Karhunen-Loeve expansions. Very heterogeneous noise is handled via collocation on a sparse grid in high dimensions. Uncertainty quantification is parallelized by coupling a multiscale mortar mixed finite element discretization with stochastic collocation. We give three new algorithms to solve the resulting system. They use the original implementation, a deterministic Multiscale Flux Basis, and a stochastic Multiscale Flux Basis. Multiscale a priori error analysis is performed and numerically verified for single-phase flow. Fourth, we present a concurrent approach that uses the Multiscale Flux Basis as an interface preconditioner. We show the preconditioner significantly reduces the number of interface iterations, and describe how it can be used for stochastic collocation as well as two-phase flow simulations in both fully-implicit and IMPES models

A Domain Decomposition Method for the Steady-State Navier-Stokes-Darcy Model with Beavers-Joseph Interface Condition

This paper proposes and analyzes a Robin-type multiphysics domain decomposition method (DDM) for the steady-state Navier-Stokes-Darcy model with three interface conditions. In addition to the two regular interface conditions for the mass conservation and the force balance, the Beavers-Joseph condition is used as the interface condition in the tangential direction. The major mathematical difficulty in adopting the Beavers-Joseph condition is that it creates an indefinite leading order contribution to the total energy budget of the system [Y. Cao et al., Comm. Math. Sci., 8 (2010), pp. 1-25; Y. Cao et al., SIAM J. Numer. Anal., 47 (2010), pp. 4239-4256]. In this paper, the well-posedness of the Navier-Stokes-Darcy model with Beavers-Joseph condition is analyzed by using a branch of nonsingular solutions. By following the idea in [Y. Cao et al., Numer. Math., 117 (2011), pp. 601-629], the three physical interface conditions are utilized together to construct the Robin-type boundary conditions on the interface and decouple the two physics which are described by Navier-Stokes and Darcy equations, respectively. Then the corresponding multiphysics DDM is proposed and analyzed. Three numerical experiments using finite elements are presented to illustrate the features of the proposed method and verify the results of the theoretical analysis

Preconditioners for Soil-Structure Interaction Problems with Significant Material Stiffness Contrast

Ph.DDOCTOR OF PHILOSOPH

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

Preconditioners for iterative solutions of large-scale linear systems arising from Biot's consolidation equations

Ph.DDOCTOR OF PHILOSOPH

State Of the Art Report in the fields of numerical analysis and scientific computing. Final version as of 16/02/2020 deliverable D4.1 of the HORIZON 2020 project EURAD.: European Joint Programme on Radioactive Waste Management

Document information Project Acronym EURAD Project Title European Joint Programme on Radioactive Waste Management Project Type European Joint Programme (EJP) EC grant agreement No. 847593 Project starting / end date 1 st June 2019-30 May 2024 Work Package No. 4 Work Package Title Development and Improvement Of NUmerical methods and Tools for modelling coupled processes Work Package Acronym DONUT Deliverable No. 4.