Bessho 



f = <^o ^s given on S, (2.7) 



then the function ^ which gives the maximum value to G is the solu- 

 tion of the Dirichlet problem, [9,10] This is easily verified by- 

 making use of the reciprocity (2»3). 



In the same way, we may construct a variational principle for 

 the Newmann problem as follows: Let us consider the extremum 

 problem for the functional 



H = iyy {<^^- 2f^)|i. dS, (2.8) 



where 



iy= '^ou is given on S. (2.9) 



This problem is seen to be equivalent to the present boundary Vcilue 

 problem by making use of (2.4), 



Alternately, we may construct a variational problem by making 

 use of (2.5); namely, by introducing the functional 



S 

 and taking the variation, we have 



J = 4yy <!>(2f^- cj>J dS, (2.10) 



6J= yy 64>(f^ - c{>J dS. (2.11) 



From this we see the equivalence to the boundary value problem. 

 [24,25] 



Now, since 



yy c|),<{>2^dS=yyy V4),V4>2 dr, (2.12) 



S D 



where D is the entire water domain and dr is a volume element, 

 a natural measure of the error of an approximate solution ^ is 



E = iyjy [V(c{>- ^,)f dr, (2.13) 



550 



