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## Über dieses Buch

Moving mesh methods are an effective, mesh-adaptation-based approach for the numerical solution of mathematical models of physical phenomena. Currently there exist three main strategies for mesh adaptation, namely, to use mesh subdivision, local high order approximation (sometimes combined with mesh subdivision), and mesh movement. The latter type of adaptive mesh method has been less well studied, both computationally and theoretically.

This book is about adaptive mesh generation and moving mesh methods for the numerical solution of time-dependent partial differential equations. It presents a general framework and theory for adaptive mesh generation and gives a comprehensive treatment of moving mesh methods and their basic components, along with their application for a number of nontrivial physical problems. Many explicit examples with computed figures illustrate the various methods and the effects of parameter choices for those methods. The partial differential equations considered are mainly parabolic (diffusion-dominated, rather than convection-dominated).

The extensive bibliography provides an invaluable guide to the literature in this field. Each chapter contains useful exercises. Graduate students, researchers and practitioners working in this area will benefit from this book.

Weizhang Huang is a Professor in the Department of Mathematics at the University of Kansas.

Robert D. Russell is a Professor in the Department of Mathematics at Simon Fraser University.

## Inhaltsverzeichnis

### Chapter 1. Introduction

In this first chapter, we introduce the basic principles of adaptivity and moving mesh methods for solving partial differential equations in one spatial dimension. In particular, two adaptive methods are described and used to solve a simple model problem - an initial-boundary value problem consisting of Burgers’ equation

1.1

$$u_t={\varepsilon}{u}_{xx}-\left(\frac{u^2}{2}\right)_x, \ \ x \in(0,1), t>0$$

subject to the boundary conditions

1.2

$$u(0,t)=u(1,t)=0$$

and initial condition

1.3

$$u(0,0)=\sin(2\pi x)+\frac{1}{2}\sin (\pi x)$$

Weizhang Huang, Robert D. Russell

### Chapter 2. Adaptive Mesh Movement in 1D

In this chapter we discuss more formally the principles of adaptive mesh movement in 1D. The underlying mesh selection problem itself is quite simple to state: If one wishes to approximate a given function u(x) using its values at a finite number of mesh points, how should these points be chosen? The answer can usually be given as follows: one chooses a so-called mesh density function ρ(x), which in some way indicates the error in the numerical approximation, and the mesh points are then placed in such a way that distances between them are smaller in regions where ρ(x) is larger, and the distances are larger in regions where ρ(x) is smaller. For the one-dimensional case, adaptivity is predicated on what is called equidistribution, which is considered in some detail here. The argument for choosing the mesh density function ρ(x) will normally be motivated by the desire to minimize an error in interpolating a function or by solving a differential equation, although in special cases other arguments such as one based on scaling invariance are used.

Weizhang Huang, Robert D. Russell

### Chapter 3. Discretization of PDEs on Time-Varying Meshes

We have seen in the previous chapters for the 1D case that the formulation of mesh movement strategies (such as for MMPDEs) is generally independent of the specific type of physical PDE being solved, and their connection to the physical solution is through the mesh density function. As well, the discretization of the physical PDE and the overall solution procedure can to a large extent be described separately from mesh movement strategies by generally assuming that a moving mesh is available. This separation is also true for multidimensional moving mesh methods. For these reasons, we assume in this chapter that a moving mesh is given and consider the problem of discretizing PDEs and overall solution procedures in the higher dimensional case. The later chapters then return to an examination of the theory and applications of multidimensional moving mesh methods.

Weizhang Huang, Robert D. Russell

### Chapter 4. Basic Principles of Multidimensional Mesh Adaptation

We have seen in Chapter 2 the crucial role that the equidistribution principle plays in designing adaptive mesh algorithms, where in fact a mesh can be fully determined from it in 1D. The situation becomes much more complicated in multidimensions. The equidistribution principle, specifying only the volume of mesh elements, is no longer sufficient for determining a multidimensional mesh. An additional condition is needed for specifying the shape and orientation of mesh elements.

Weizhang Huang, Robert D. Russell

### Chapter 5. Monitor Functions

From the previous chapter we have seen that an adaptive mesh can conveniently be viewed as an

M

–uniform mesh, or a uniform one in a metric space equipped with metric tensor or monitor function

M

=

M(x)

. A key to the success to this approach of mesh adaptation is in the selection of a proper M. In this chapter, we shall study how to define the monitor function based on estimates for interpolation error, (semi-) a posteriori error bounds for the solution, or some other geometric and physical considerations. For a given monitor function M, the central issue for mesh adaptation then becomes generating an M-uniform mesh. This issue is addressed in Chapters 6

Weizhang Huang, Robert D. Russell

### Chapter 6. Variational Mesh Adaptation Methods

In this chapter and the following one, we discuss the general mesh generation problem. The first main class of methods we consider are variational methods. They are applicable for either nonadaptive or adaptive mesh generation, and natural relationships between these two different goals are examined. While mesh generation ideas generally apply for either the static or dynamic case, we often limit discussion to static mesh generation since extending it to compute a dynamic mesh is straightforward in principle using the MMPDE approach discussed in § 6.1.2 (see also see § 2.3). Although variational methods have most commonly been used for finite difference computations for structured meshes, they can also be employed for unstructured mesh generation and adaptation (e.g., see Cao, Huang, and Russell [81]) and for mesh smoothing (e.g., see Canann et al. [79] and Knupp [215]).

Weizhang Huang, Robert D. Russell

### Chapter 7. Velocity-Based Adaptive Methods

In this chapter we discuss

velocity-based

adaptive moving mesh methods. Although the classification of methods as being either velocity-based or location-based can at times be somewhat artificial, the former are generally characterized by the fact that their formulations directly target the mesh velocity, with the subsequent mesh points determined by integrating the velocity field. Some of these methods are motivated by the Lagrangian method in computational fluid dynamics (e.g., see Fletcher [148] or §7.1.1) and some others are based on minimizing a quantity related to error. A fortuitous property of the Lagrangian methods is that it is well-suited to maintaining sharper material interfaces since convection terms are eliminated from the governing equations. A disadvantage is that the meshes have a tendency to tangle and lose spatial resolution of the solution. Unfortunately, the Lagrangian-like moving mesh methods also inherit this disadvantage of Lagrangian methods, and major effort has gone into the development of these methods so as to avoid mesh tangling and/or regain spatial accuracy.

Weizhang Huang, Robert D. Russell

### Backmatter

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