Numerical Solution of Boundary-Value Problems
In this chapter we describe and formulate an algorithm embodying the method of difference potentials for the numerical solution of the boundary-value problems $$ \Delta u_{\overline D } = 0,x \in D,u_{\overline D } \left| {_\Gamma = \phi (s)} \right., $$
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In this chapt er we describe and formul ate an algorithm embodying the method of difference potentials for the numerical solution of the boundaryvalue problems LlUD = 0 ,
uDlr = ep(s),
XE D ,
and LlUD = 0,
X E D,
(UD
+
8;: )Ir
(I)
(II)
= ep(s).
These boundary-value problems were reduced in Chap. 3 to the following problems on t he boundary: uj- -
Prvr
= 0,
UDlr = ep(s),
an d ( UD +
ur - Prur = 0,
8;:)Ir
(III)
(IV)
= ep(s),
resp ectively. Let us writ e the boundar y-value problems (III) and (IV) in the unified form ur - Prur
= 0,
lur
=
ep(s),
Ur E Ur .
ip E
Pr ,
(V)
at taching different meanings to lUr , nam ely,
lur = {
for problem (III),
u Dlr (UD
+
D) O;n
Ir
for problem (IV).
4.1 Intermediate Discretization 4.1.1 The Finite-Dimensional Variational Problem Here we replace problem (V) by the approximat e problem of finding the minimum of a quadratic functi on of finitely many arguments. This variatio nal problem will serve as a st arting point for the const ru ct ion of another V. S. Ryaben'kii, Method of Difference Potentials and Its Applications © Springer-Verlag Berlin Heidelberg 2002
88
4. Numerical Solution of Boundary-Value Problems
finite-dimensional problem that will be used in the following section for the numerical solution of problem (V). We begin by approximating the spaces Ur and Pr (appearing in the statement of problem (V)) by finite-dimensional spaces. To be definite, the length of the boundary T of the domain [) is assumed to be equal to unity. Suppose that N is a given positive integer , and let us mark equidistant points along the boundary T by Sj = jH, j = 0,1, ... , N -1, Nh = 1. Let w = w(H) be the set of these points. Consider the Sobolev space W~w of scalar functions v w defined on the grid w, with norm k = 0,1,2 ....
(4.1)
Consider also the space Uk,w = W~w = W~w EB W;,:l of vector functions v W = (v(O) v(1») w' w '
v(O) w
IIvwllk,w =
E W2k,w '
IIv~O)llk,w
v(1) W
k- 1 E W2,w ,
(4.2)
+ IIv~l)lIk_l,w .
Let p be a positive integer, and consider the operation R~~p) : Uw ---+ Ur which assigns to each Uw = (u~), uS1») E Uw a certain vector function
Here
R(H,p) T'i»
'
where
R(H,p)v Ft»
w
= v(H,p) r , is the operation of smooth completion
of a scalar function V w defined on the set w to the function vW,p) defined everywhere on r and having continuous derivatives up to order p inclusive, using the local smooth completion formulas (Sect. 1.1). Consider the boundary-value problem (V): ur-Prur =0,
lur = cp,
(4.3)
where by l: Ur ---+ P r we shall mean, as above, different operators for problems (III) and (IV). We shall seek an approximate solution of (4.3) as u(H,p) -
r
-
R(H,p)u Tea w·
Substituting this expression into (4.3), we obtain the following equations for 2N unknown numbers forming an element Uw E Uw : (4.4) Obviously, the system (4.4) is overdetermined and, generally speaking, cannot be satisfied for any choice of U w E Uw ' Let us define the generalized solution
4.1 Intermediate Discretization
89
of (4.4) in terms of the
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