1.1 TYPES OF PDEs

As we will see in this book, elliptic boundary value problems are the type of problem to which multigrid methods can be applied very efficiently. However, multigrid or multigrid-like methods have also been developed for many PDEs with nonelliptic features.

We will start with the usual classification of second-order scalar 2D PDEs. Generalizations of this classification to 3D, higher order equations or systems of PDEs can be found [150]. We consider equations Lu = f in some domain where

     (1.1.1)

with coefficients aij, ai, ao and a right-hand side f which, in general, may depend on x, y, u, ux, uy (the quasilinear case). In most parts of this book, Lu = f is assumed to be a linear differential equation, which means that the coefficients and the right-hand side f only depend on (x, y). L is called

In general, this classification depends on (x, y) and, in the nonlinear case, also on the solution u. Prototypes of the above equation are

Since multigrid methods work excellently for nicely elliptic problems, most of our presentation in the first chapters is oriented to Poisson’s and Poisson-like equations. Other important model equations that we will treat in this book include

All these equations will serve as model equations for special features and complications and are thus representative of a larger class of problems with similar features. These model equations depend crucially on a parameter ε or τ. For certain parameter values we have a singular perturbation: the type of the equation changes and the solution behaves qualitatively different (if it exists at all). For instance, the anisotropic equation becomes parabolic for ε → 0, the equation with mixed derivatives is elliptic for |τ| < 2, parabolic for |τ| = 2 and hyperbolic for |τ| > 2. All the model equations represent classes of problems which are of practical relevance.

In this book, the applicability of multigrid is connected to a quantity, the “h-ellipticity measure Eh“, that we will introduce in Section 4.7. This h-ellipticity measure is not applied to the differential operator itself, but to the corresponding discrete operator. It can be used to analyze whether or not the discretization is appropriate for a straightforward multigrid treatment. Nonelliptic problems can also have some h-ellipticity if discretized accordingly.

The above model equations, except the wave and the heat conduction equations, are typically connected with pure boundary conditions. The wave and the heat conduction equations are typically characterized by initial conditions with respect to one of the variables (y) and by boundary conditions with respect to the other (x).

We will call problems which are characterized by pure boundary conditions space-type problems. For problems with initial conditions, we interpret the variable for which the initial condition is stated as the time variable t (= y), and call these problems time-type. Usually these problems exhibit a marching or evolution behavior with respect to t. Space-type equations, on the other hand, usually describe stationary situations. Note that this distinction is different from the standard classification of elliptic, hyperbolic and parabolic. Elliptic problems are usually space-type while hyperbolic and parabolic problems are often time-type. However, the stationary supersonic full potential equation, the convection equation (see Chapter 7) and the stationary Euler equations (see Section 8.9) are examples of hyperbolic equations with space-type behavior. (In certain situations, the same equation can be interpreted as space- or as time-type, and each interpretation may have its specific meaning. An example is the convection equation.)

For time-type problems, there is usually the option to discretize the time variable explicitly, implicitly or in a hybrid manner (i.e. semi-implicitly). The implicit or semi-implicit discretization yields discrete space-type problems which have to be solved in each time step. We will briefly consider multigrid approaches for the solution of such space-type problems which arise in each time step in Section 2.8. Typically, these problems have similar but more convenient numerical properties than the corresponding stationary problems with respect to multigrid.

1.2 GRIDS AND DISCRETIZATION APPROACHES

In this book, we assume that the differential equations to be solved are discretized on a suitable grid (or, synonymously, mesh). Here, we give a rough survey (with some examples) of those types of grids which are treated systematically in this book and those which are only touched on. We also make some remarks about discretization approaches.