ABSTRACT
This book provides the required elements for applying modern techniques to the discretization of a wide range of continuum mechanics problems. Mimetic or compatible methods begin by discretizing the vector calculus operators used to describe continuous problems. They then use the resulting discrete operators to discretize the given problem. If the continuous problem is stable, the mimetic discretization will typically result in a stable numerical scheme. However, although stability is a desirable characteristic of any numerical method, it is usually very dicult to accomplish, especially in regard to high-order discretization. Many classical discretization approaches begin by discretizing a given problem to some specied order of accuracy, followed by an attempt to prove the method's stability, which can be a very dicult task. In contrast, mimetic methods automatically preserve many important properties of the continuous problem (e.g., conservation laws), thereby contributing to the stability of the method. Continuum models are used in many sciences, and their relevant mag-
nitudes are usually described as unknown distributions or elds, which are dependent upon time and space in the continuous domain under consideration. If such elds are piecewise continuous or locally integrable dependent variables, then the conservation laws of the continuum models can be expressed using the integral forms of the underlying conservation principle. Mimetic methods are used in general logically rectangular grids [2], [3];
irregular or unstructured grids [4], [5], [6], [7]; triangular grids [8], [9], [10]; and polygonal grids [11], [12], [13]. They can be made higher order in 3-D [14], as well as monotone [15]. Mimetic spectral and pseudo-spectral methods have also been derived [16], [17], [18], and have been implemented in a multiscale setting [19] using interpolation [20]. Iterative solvers have been used to invert the matrices arising from mimetic discretization [21]. There are many applications of mimetic methods in solving continuum problems, including in the geosciences (porous media) [19], [22], [23], [24], [25], [26], [27], [28]; uid dynamics (Navier-Stokes, shallow water) [29], [30], [31]; image processing [32], [33], [34], [35]; general relativity [2]; and electromagnetism [36]. Mathematical models of continuum mechanics problems are typically de-
scribed by boundary value problems, and expressed as either a system of partial dierential equations (PDEs) or as integral equations. To facilitate their numerical solution, these equations can be discretized by any one of a large number of techniques; standard methods include various nite dierence
and nite element approaches. These traditional methods are often applied by discretizing the dening system of equations directly. One disadvantage of such an approach is that the discretization scheme selected may have little connection with the underlying physical problem. Mimetic methods, on the other hand, begin by rst discretizing a problem's underlying continuum theory. By \discretizing the continuum theory," we mean that mimetic methods initially construct a discrete mathematical analog of a relevant description of continuum mechanics (with the description usually taking the form of a physical conservation or constitutive law). The discrete form of these conservation or constitutive law constrains the structure that discrete operators can take. After building discrete operators that obey the discretized physical law, these mimetic operators can then be substituted into a system of partial dierential equations or integral equations. This yields a mimetic discretization for the boundary value problem, which automatically satises the discrete version of the physical law on the corresponding domain under consideration. As a result, discretizations obtained using mimetic methods tend to replicate much of the behavior found in the actual continuum problem. Since the physical laws are, in eect, built into the discretization, mimetic methods turn out to be good candidates for modeling even the most challenging problems, such as those involving anisotropic or strongly inhomogeneous material properties. The physical basis for mimetic discretizations also helps to reduce the oc-
currence of various nonphysical numerical artifacts that can occur when using a traditional discretization technique. Examples of such artifacts include
1. Oscillatory solution contaminants (also known as hourglass modes);
2. Nonphysical spectra for Laplacian operators, when dealing with elliptic partial dierential equations; and
3. Late-time instabilities in which errors increase rapidly and signicantly over time.
