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Über dieses Buch

This book offers a thorough guide starting from fundamental functional analysis leading to the coupling of Stokes and Darcy equations, including numerical analysis and scientific computing. Almost all intermediate results are given with complete, rigorous proofs, including theorems which can be rarely found in the literature such that this book serves well as a reference on the topic. Special care is taken to analyze the difficult cases of non-smooth interfaces which are not completely enclosed in one subdomain, i.e, intersect with the outer boundary. This can hardly be found in the literature. Additionally, known and new subdomain iterative methods are introduced, analyzed and applied to standard examples as well as one example motivated by a geoscientific setting.



Chapter 1. Introduction

Flows in domains which are partly occupied by a porous medium are of great interest and importance, noticeable examples include groundwater—surface water flow, as well as air and oil filters, blood filtration through vessel walls, and fuel cells.
Ulrich Wilbrandt

Chapter 2. Notation and Preliminary Results

In this chapter most of the notation used in this monograph is introduced; in particular, Lipschitz domains on which the so-called Lebesgue and Sobolev spaces are defined, together with a few basic inequalities. Furthermore, the important theorem of Lax–Milgram is shown. To begin with, some definitions and results from functional analysis are stated.
Ulrich Wilbrandt

Chapter 3. Properties of Sobolev Spaces

This chapter collects a number of important properties of Sobolev spaces. Almost every claim is provided together with a proof. The main result is the density of \(\mathcal D(\overline \varOmega )\) in W s, p(Ω) and proved in Theorem 3.4.5. The necessary tools to establish these proofs are introduced and intermediate results are presented in the following subsections. This entire chapter can be viewed as a preparation for subsequent ones on traces, Chap. 4, and to meaningfully define the weak forms of some partial differential equations in Chap. 5.
Ulrich Wilbrandt

Chapter 4. Traces

One important property of certain Sobolev spaces is the fact that there is a well defined restriction onto the boundary, even though the boundary has measure zero. Such restrictions are known as traces and allow for prescribed boundary data of solutions of partial differential equations.
Ulrich Wilbrandt

Chapter 5. Subproblems Individually

In this section the existence and uniqueness of solutions of the involved subproblems are proved. Besides the Laplace equation this includes the so-called saddle point theory which is an abstract framework and is then applied to the Stokes equations as well as to the coupled Stokes–Darcy problem in Chap. 6.
Ulrich Wilbrandt

Chapter 6. Stokes–Darcy Equations

Let \(\varOmega \subset \mathbb {R}^d\) be a Lipschitz domain split into two disjoint nonempty subdomains Ω p and Ω f which are Lipschitz, too. The index p refers to the Darcy subdomain where a porous medium is modeled, while the index f refers to the free flow domain with a Stokes model.
Ulrich Wilbrandt

Chapter 7. Algorithms

The Neumann–Neumann as well as the Robin–Robin systems (6.​7) and (6.​34) along with their decoupled variants (6.​20) and (6.​40) can be solved iteratively.
Ulrich Wilbrandt

Chapter 8. Numerical Results

The algorithms described in Chap. 7 are implemented in the C++ finite element code ParMooN (Wilbrandt et al., Comput Math Appl 74(1):74–88, 2017). In this chapter several examples from the literature are introduced and numerical results shown. To begin with, a more general discussion on numerical examples is given.
Ulrich Wilbrandt


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