There are a number of alternatives that may help surmount the diculties described in earlier sections.
3.6.1 Improvements to nite dierences
There are a few directions in which nite dierence methods may be improved. One is to develop better algorithms for solving large sparse systems of linear equations, so that the
unstable coecients generated by nite dierence techniques using irregular sample points would become solvable.
A distinctly dierent approach would be to simply do nite dierences on regular grids, and to basically follow the Chesshire-Henshaw idea. While their idea works well for some special problems, however, there are cases when their idea produces less reasonable answers.
For a discussion of this, see [23].
3.6.2 Improvements to nite elements
To improve the performance of nite element methods on manifolds, on the other hand, probably requires more work. While FEMs work admirably well with irregular sampling geometry, the complexity of the geometric problem of combining local equations into a global system can be rather daunting, as was shown in this section. Clearly, much more work needs to be done in this domain, and there are many variations on these ideas. Part of the diculty of this problem is that, in view of the variational formulation of Laplace's equation, the problem of combining local equations is that of a constrained minimization problem, which are often non-trivial. On the other hand, perhaps a standard technique like Lagrange multipliers would work nicely for this case. There are many other things to try.
On the other hand, one of the diculties that arises with the semi-local method is that it gives charts little control over the geometry of their local meshes because nodes are copied between charts. Thus, while the method produces reasonably good results and has nice convergence properties, it does accumulate quite a bit of truncation error due to geometric defects. It would be very useful to generalize the idea in a way that still allows regular local grids, so as to minimize the eects of geometry on accuracy.
3.6.3 Other methods
Finally, there could be breakthroughs in mesh generation on arbitrary
n
-manifolds. Al-though most current work have focused on low-dimensional problems because of their po-tential applications in engineering and computer graphics, this is a rather active research area and much is being discovered. A global nite element method should work rather nicely on a manifold.Or one could exploit the meshless methods developed by Duarte and Oden [11], which explicitly build partitions of unity using discrete sample points without rst generating a mesh. This has the advantage that one does not need to think about combining meshes to use these methods on manifolds. Furthermore, their method can utilize essentially Rayleigh-Ritz or Galerkin approximations, so that the resulting linear equations are solvable by iterative methods.
Hyperbolic equations
This topic of this chapter is the numerical solution of partial dierential equations that describe how certain physical systems evolve in time. Again, as in the solution of ellip-tic boundary value problems on manifolds, it is possible to break this problem into two components: First, we must have a way of locally integrating the PDE; and second, the local solutions must be combined to form a global solution. It is also possible, of course, to discretize the entire manifold rst before solving the equations, but it will turn out that the diculties one must overcome in global methods are not all that dierent from those of local methods. Because of the nontrivial nature of solving such equations even in the case where the domain has trivial geometry, this chapter focuses on the local problem.
Standard PDE solvers generally perform nite element or nite dierence approxima-tions in space rst, so as to compute the time derivative, and then step forward uniformly in timeat regular intervals|As one would with ordinary dierential equations.1 While this approach works well enough for many problems, it is rather unsatisfactory philosophically:
We have good reason to believe that physical reality does not distinguish among time-like directions, and that any time axis is just as fundamental and just as arbitrary as any other. Thus, a coordinate-independent description of fundamental physical processes and the equations that govern them should not depend on the existence of a unique time axis.
More pragmatically, there exist physical problems for which it is helpful to use dierent frames of reference, and a properly coordinate-independent formulation of PDEs should not be restricted to advancing along an arbitrarily chosen time axis. The use of regular time steps implicitly gives the time coordinate a special status, which complicates any attempt at coordinate-independent representations and solutions.
1A notable exception occurs in numerical general relativity, where the use of Regge calculus suggests some interesting ideas for the work at hand. Einstein's eld equations are very much beyond the scope of this report, though, and will not be discussed here. For more information on Regge calculus, see Sorkin [26].
For a good introduction to general relativity, see Schutz [25].
One natural solution to this dilemma is the following: Instead of discretizing the spatial dimensions and stepping forward in time, one simply discretizes the equation over spacetime2 and solve for the unknown solution over the entire spacetime region of interest in one step.
One might expect, for example, that standard nite element techniques may be applied directly to the entire spacetime domain, and that the unknown solution can be solved over all spacetime events by solving one very large system of algebraic equations.
Perhaps not too surprisingly, this simple idea does not work, even though there are no obvious problems in the derivation. One reason for this failure is proposed in the next section, and, in view of this proposal, various ways for improving the accuracy are suggested in x4.3. x4.4 discusses some of the diculties that arise in these improved methods, and also presents some problems that spacetime methods must, in general, overcome. Finally, possible directions for future research in this area are suggested inx4.5.
This chapter is more about open questions than solutions to well-posed problems, and as such may be seem less coherent than earlier chapters. However, it is hoped that the questions asked here will lead to other questions whose answers will some day shed light on the mathematical, physical, and computational structures involved in understanding partial dierential equations. Also, because everything here is performed in subsets of Euclidean space, explicit programs probably do not aid in understanding, and are thus omitted in this chapter.
As in earlier chapters, the focus here will be on the simplest possible example that exhibits interesting behavior, which in this case is the linear wave equation.
4.1 The linear wave equation
While Laplace's equation is arguably one of the most important PDEs, there are other important equations that have fundamentally dierent behavior. One of these is the linear wave equation. This equation describes, for example, the propagation of electromagnetic waves in free space. It is therefore useful to identify one of the variables as time in some frame of reference, and to dene
D
t=D
n+1so that time and space derivatives can be more easily distinguished. The wave equation in (n
+ 1) dimensions (n
space dimensions plus time) is then:(
D
t2;c
2)u
= 0;
(4.1)where
c >
0 is a real constant and = r2 =Pni=1D
i2 is the Laplacian operator over the space variables. For concreteness, this discussion will be restricted to the casen
= 1. In2Spacetime is simply the set of all spatial positions of our space along with time indices. Points in spacetime are often calledevents, and Figure 4-1 would be an example of aspacetime diagram.
this case, the wave equation also describes the behavior of a vibrating string with small oscillations. For convenience, let us dene
D
x =D
1 so that =D
x2.In constrast to Laplace's equation, the boundary value problem for the wave equation is ill-posed. That is, it does not always have solutions for arbitrary boundary conditions, and even when such solutions exist, they are often not unique. However, in the case when
n
= 1 and is the unit square f(x;t
) 2R
2j0x
1;
0t
1g, one can specify initial conditionsu
(x;
0) =f
(x
); D
tu
(x;
0) =g
(x
);
(4.2)u
(0;t
) =h
(t
); u
(1;t
) =k
(t
);
(4.3) for some prescribed functionsf
,g
,h
, andk
. Then the wave equation does have a unique solution. This is called the initial value problem.3It is tempting to apply the nite element method directly to the initial value problem for the wave equation. In particular, Galerkin's method may seem generally applicable.
However, there is good evidence that Galerkin's method, as presented in Appendix A, will almost always do poorly for the linear wave equation. This does not, of course, imply that nite element methods cannot be somehow adapted for the wave equation. First, though, let us take a closer look at why boundary value problems are ill-posed for the linear wave equation.