Zienkiewicz O.C., Taylor R.L. Vol. 3. The finite - tiera.ru

with inaccuracies and instabilities in the solution which follow the arbitrary use of this
weighting function.
This chapter will discuss the manner in which these di culties can be overcome and
the approximation improved.
We shall in the main address the problem of solving Eq. (2.4), i.e. the scalar form,
and to simplify matters further we shall often start with the idealized one-dimensional
equation:
@ @ @
‡ U ÿ
@t @x @x
k @
@x
‡ Q ˆ 0 …2:10†
The term @U=@x has been removed here for simplicity. The above reduces in steady
state to an ordinary di€erential equation:
U d d
ÿ
dx dx
k d
dx
‡ Q ˆ 0 …2:11†
in which we shall often assume U, k and Q to be constant. The basic concepts will be
evident from the above which will later be extended to multidimensional problems,
still treating as a scalar variable.
Indeed the methodology of dealing with the ®rst space derivatives occurring in
di€erential equations governing a problem, which as shown in Chapter 3 of
Volume 1 lead to non-self-adjointness, opens the way for many new physical
situations.
The present chapter will be divided into three parts. Part I deals with steady-state
situations starting from Eq. (2.11), Part II with transient solutions starting from Eq.
(2.10) and Part III dealing with vector-valued functions. Although the scalar problem
will mainly be dealt with here in detail, the discussion of the procedures can indicate
the choice of optimal ones which will have much bearing on the solution of the general
case of Eq. (2.1). We shall only discuss brie¯y the extension of some procedures to the
vector case in Part III as such extensions are generally heuristic.
Part I: Steady state
2.2 The steady-state problem in one dimension
2.2.1 Some preliminaries
The steady-state problem in one dimension 15
We shall consider the discretization of Eq. (2.11) with
X Ni ~ i ˆ N ~ f …2:12†
where N k are shape functions and ~ f represents a set of still unknown parameters.
Here we shall take these to be the nodal values of . This gives for a typical internal
node i the approximating equation
K ij ~ j ‡ f i ˆ 0 …2:13†