13 The finite element method for partial differential equations
13.1 Introduction
When we consider a partial differential equation, such as the ubiquitous Laplace
equation
∇^2 φ(r)=0, (13.1)together with some boundary condition(s), the obvious way of solving it that comes
to mind is to discretise this equation on a regular grid, hoping that this grid can
match the boundary in some way. Then we solve the discretised problem using,
for example, iterative methods such as the Gauss–Seidel or conjugate gradients
method (see Appendix A7.2). For many problems, this approach is adequate, but
if the problem is difficult in the sense that it has a lot of structure on small scales
in some region of the domain, or if the boundary has a complicated shape which
is difficult to match with a regular grid, it might be useful to apply methods that
allow for flexibility of the grid on which the solution is formulated. In this chapter
we discuss such a method, thefinite element method.
One way of looking at the finite element method (FEM) is by realising that many
partial differential equations can be viewed as solution methods for variational
problems. In the case of the Laplace equation with zero boundary condition, for
example, finding the stationary solution of the functional
∫
D[∇φ(r)]^2 ddr, (13.2)where the integral is over thed-dimensional domainDand where we confine
ourselves to functionsφ(r)which vanish on the domain boundary, yields the same
solution as that of the Laplace equation – in fact, the Laplace equation is the Euler
equation for this functional (see the next section).
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