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2.3 Forcing Functiontex2html_wrap_inline6769


For the last two-dimensional experiment, the problem is now acting two different two-variable polynomials. The input data has a jumped discontinuous property at the junction. We consider the following problem

eqnarray2530

Using the same technique of finding the exact solution as the previous experiment and the integration by parts, the coefficient of tex2html_wrap_inline6301 can be determined as follows

displaymath6773

Therefore, the exact solution for the problem (2.3.1) is

eqnarray2554

Again, we used the five-point difference method to solve the numerical solutions and setting the function value at the junctions to be the average of the function values neighbouring nodal points. Table 8.1 and Table 8.2 show the numerical solutions at the nodal point (1/4, 1/4) with zero at the junctions and Table 8.3 and Table 8.4 show the numerical solutions at the same point with the average at the junctions. And it is obvious that the numerical results strongly agreed with the results of the Experiment (2.2). We also obtain the same asymptotic error expansion with order tex2html_wrap_inline6777 when we set the average at the junctions.


 

TABLE 8.1
 
h=1/4
-.18078346x10-2
-.59652469x10-3
-.46915394x10-4
.23444242x10-6
h=1/8
-.12021797x10-2
-.18431772x10-3
-.56592872x10-5
.58813053x10-7
h=1/16
-.69324869x10-3
-.50323895x10-4
-.65594948x10-6
.54209933x10-8
h=1/32
-.37178629x10-3
-.13072936x10-4
-.77250315x10-7
.39677357x10-9
h=1/64
-.19242961x10-3
-.33261717x10-5
-.93091126x10-8
  
h=1/128
-.97877893x10-4
-.83852476x10-6
    
h=1/256
-.49358209x10-4
      


 

TABLE 8.2
r(1)
.66498320
.30898590
.12062751
.25086353
r(2)
.57665981
.27302798
.11590673
.092173303
r(3)
.53629570
.25977591
.11776870
.073192041
r(4)
.51758125
.25443188
.12050582
  
r(5)
.50864257
.25209906
    
r(6)
.50428352
      


 
 

TABLE 8.3
 
h=1/4
.32403618x10-3
.18540080x10-4
.20150793x10-6
.36028947x10-9
h=1/8
.94914105x10-4
.13476687x10-5
.35032214x10-8
.33401684x10-11
h=1/16
.24739278x10-4
.87513561x10-7
.58025812x10-10
.65175122x10-12
h=1/32
.62504546x10-5
.55239968x10-8
.15482208x10-11
.63103299x10-12
h=1/64
.15667567x10-5
.34670126x10-9
.64536397x10-12
  
h=1/128
.39194919x10-6
.22273857x10-10
    
h=1/256
.98004003x10-7
      

 

TABLE 8.4
r(1)
.29291206
.072689475
.017385030
.0092707912
r(2)
.26064912
.064937001
.016563558
  
r(3)
.25265308
.063121609
.026681587
  
r(4)
.25066283
.062762755
    
r(5)
.25016596
.064245101
    
r(6)
.25004262
      

 

Cheung Sau Hung

Wed Sep 15 10:03:39 HKT 1999