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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 24 "Poisson's Equation in 2D
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "with(linalg):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "w
ith(plots):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected nam
es norm and trace have been redefined and unprotected\n" }}{PARA 7 "" 
1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n
" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 17 "Analytic Solution" }}{PARA 
0 "" 0 "" {TEXT -1 47 "(see the lectures about obtaining the solution)
" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "We consider the solution" }
{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "u_an := (x,y) -
> sin(Pi*x)*sinh(Pi*y)/sinh(Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
%u_anGR6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF)*&*&-%$sinG6#*&%#PiG\"\"
\"9$F4F4-%%sinhG6#*&F3F49%F4F4F4-F76#F3!\"\"F)F)F)" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 67 "which solves the Poisson equation with homogeneou
s right hand side:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 43 "-diff(u_an(x,y),x,x) - diff(u_an(x,y),y,y);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "and satis
fies the following boundary conditions:" }{MPLTEXT 1 0 0 "" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 21 "u_an(0,y); u_an(1,y);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "u_an(x,0); u_an(x,1);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$sinG6#*&
%#PiG\"\"\"%\"xGF(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Here's a pl
ot of the solution:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 34 "plot3d(u_an,0..1,0..1,axes=boxed);" }}{PARA 13 "" 1 "" 
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2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}}{SECT 0 {PARA 3 "" 
0 "" {TEXT -1 14 "Discretization" }}{PARA 0 "" 0 "" {TEXT -1 212 "We c
onsider a two-dimensional mesh of grid points. We will prescribe the t
emperature at the boundary points (Dirichlet conditions), and compute \+
the values at the inner points via the discretized poisson equation." 
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "N and M denote the number of gri
d intervals in x- and y-direction:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
15 "N := 4; M := 4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"\"%" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG\"\"%" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 59 "The arrays x and y hold the coordinates of the grid poi
nts:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "x := arr
ay(0..N, [seq(n/N, n=0..N)]): \ny := array(0..M, [seq(m/M, m=0..M)]):
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "And the " }{XPPEDIT 18 0 "(N+1
)*(M+1);" "6#*&,&%\"NG\"\"\"F&F&F&,&%\"MGF&F&F&F&" }{TEXT -1 10 " unkn
owns " }{XPPEDIT 18 0 "u[n,m];" "6#&%\"uG6$%\"nG%\"mG" }{TEXT -1 19 " \+
will be stored in " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "unbek := seq(
seq(u[n,m], n=0..N), m=0..M);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&un
bekG6;&%\"uG6$\"\"!F)&F'6$\"\"\"F)&F'6$\"\"#F)&F'6$\"\"$F)&F'6$\"\"%F)
&F'6$F)F,&F'6$F,F,&F'6$F/F,&F'6$F2F,&F'6$F5F,&F'6$F)F/&F'6$F,F/&F'6$F/
F/&F'6$F2F/&F'6$F5F/&F'6$F)F2&F'6$F,F2&F'6$F/F2&F'6$F2F2&F'6$F5F2&F'6$
F)F5&F'6$F,F5&F'6$F/F5&F'6$F2F5&F'6$F5F5" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 60 "The function f will be used to compute the right hand sid
es:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "f := (x,y
) -> 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG\"\"!" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 53 "And the function g will implement the bou
ndary values" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 115 
"g := proc(x,y)\n   if x=0 or x=1 or y=0 then return 0\n   else if y=1
 then return sin(Pi*x) end if; end if;\nend proc;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"gGR6$%\"xG%\"yG6\"F)F)@%55/9$\"\"!/F.\"\"\"/9%F/OF/
@$/F3F1O-%$sinG6#*&%#PiGF1F.F1F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 53 "Und unser Gleichungssystem sieht damit wie folgt aus:" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 313 "gls := \{ seq( seq(\n            -u[n-1,
m]-u[n,m-1] + 4*u[n,m] \n            -u[n+1,m]-u[n,m+1] = f(x[n],y[n])
,\n         n=1..N-1), m=1..M-1),\n         seq( u[n,0]=g(x[n],0), n=0
..M ), \n         seq( u[n,M]=g(x[n],1), n=0..M ),\n         seq( u[0,
m]=g(0,y[m]), m=1..M-1), \n         seq( u[N,m]=g(1,y[m]), m=1..M-1) \+
\};" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$glsG<;/,,&%\"uG6$\"\"!\"\"\"
!\"\"&F)6$F,F+F-*&\"\"%F,&F)6$F,F,F,F,&F)6$\"\"#F,F-&F)6$F,F6F-F+/,,F2
F-&F)6$F6F+F-*&F1F,F4F,F,&F)6$\"\"$F,F-&F)6$F6F6F-F+/,,F4F-&F)6$F@F+F-
*&F1F,F>F,F,&F)6$F1F,F-&F)6$F@F6F-F+/,,&F)6$F+F6F-F2F-*&F1F,F7F,F,FAF-
&F)6$F,F@F-F+/,,F7F-F4F-*&F1F,FAF,F,FJF-&F)6$F6F@F-F+/,,FAF-F>F-*&F1F,
FJF,F,&F)6$F1F6F-&F)6$F@F@F-F+/,,&F)6$F+F@F-F7F-*&F1F,FQF,F,FVF-&F)6$F
,F1F-F+/,,FQF-FAF-*&F1F,FVF,F,FgnF-&F)6$F6F1F-F+/,,FVF-FJF-*&F1F,FgnF,
F,&F)6$F1F@F-&F)6$F@F1F-F+/&F)6$F+F+F+/F.F+/F;F+/FEF+/&F)6$F1F+F+/&F)6
$F+F1F+/F^o,$*$-%%sqrtG6#F6F,#F,F6/FcoF,/FjoFip/&F)6$F1F1F+/F(F+/FNF+/
F[oF+/FHF+/FenF+/FhoF+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "l
sg := solve(gls,\{unbek\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$lsgG
<;/&%\"uG6$\"\"!F*F*/&F(6$\"\"\"F*F*/&F(6$\"\"#F*F*/&F(6$\"\"$F*F*/&F(
6$\"\"%F*F*/&F(6$F*F:F*/&F(6$F.F:,$*$-%%sqrtG6#F2F.#F.F2/&F(6$F2F:F./&
F(6$F6F:FA/&F(6$F:F:F*/&F(6$F*F.F*/&F(6$F*F2F*/&F(6$F*F6F*/&F(6$F:F.F*
/&F(6$F:F2F*/&F(6$F:F6F*/&F(6$F6F6,&#\"#6\"$7\"F.*&#\"#P\"$C#F.FCF.F./
&F(6$F2F6,&#FeoFboF.*&F`oF.FCF.F./&F(6$F.F6F_o/&F(6$F2F2,&#F.\"\")F.*&
#F.\"#;F.FCF.F./&F(6$F6F2,&FgpF.*&FgpF.FCF.F./&F(6$F.F2F\\q/&F(6$F6F.,
&#F6FboF.*&#\"\"&FfoF.FCF.F./&F(6$F.F.Fdq/&F(6$F2F.,&#FhqFboF.*&FeqF.F
CF.F." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 21 "Plotting the solution" 
}}{PARA 0 "" 0 "" {TEXT -1 93 "Maple provides a function surfplot that
 can produce a 3D function plot out of discrete data. " }}{PARA 0 "" 
0 "" {TEXT -1 68 "However, it requires the solution to be present as a
 list of points." }}{PARA 0 "" 0 "" {TEXT -1 59 "We'll define a couple
 of helper functions for that purpose:" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 104 "The first function produces one point of the solution (i
.e. a triple of x-coord., y-coord. , and value):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 48 "punkt := (i,j) -> [x[i],y[j],subs(lsg, u[i,j])];" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&punktGR6$%\"iG%\"jG6\"6$%)operatorG
%&arrowGF)7%&%\"xG6#9$&%\"yG6#9%-%%subsG6$%$lsgG&%\"uG6$F1F5F)F)F)" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "See the midpoint as an example:" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "punkt(N/2, M/2);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#7%#\"\"\"\"\"#F$,&#F%\"\")F%*&#F%\"#;F%-%%sqrtG6#F
&F%F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 141 "Now, surfdata can do it
s work, and produce a nice plot. Note that it interpolates the data, w
hich may lead to certain artefacts in the plot. " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 78 "surfdata([seq([seq(punkt(i,j), i=0..N)],\n           \+
    j=0..N)],axes=BOXED); " }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 
{PLOTDATA 3 "6$-%%MESHG6#7'7'7%$\"\"!F*F)F)7%$\"+++++D!#5F)F)7%$\"++++
+]F.F)F)7%$\"+++++vF.F)F)7%$\"\"\"F*F)F)7'7%F)F,F)7%F,F,$\"+I\")HNe!#6
7%F0F,$\"+cxN_#)F=7%F3F,F;7%F6F,F)7'7%F)F0F)7%F,F0$\"+wM))3:F.7%F0F0$
\"+wM)Q8#F.7%F3F0FF7%F6F0F)7'7%F)F3F)7%F,F3$\"+:17=LF.7%F0F3$\"+x6`#p%
F.7%F3F3FP7%F6F3F)7'7%F)F6F)7%F,F6$\"+5y1rqF.7%F0F6F67%F3F6FZ7%F6F6F)-
%*AXESSTYLEG6#%$BOXG" 1 2 0 1 10 0 2 1 1 2 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "In a s
imilar way, we can produce an error plot." }{MPLTEXT 1 0 0 "" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 65 "fehler := (i,j) -> [x[i],y[j],subs(lsg, u
[i,j])-u_an(x[i],y[j])];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "surfdat
a([seq([seq(fehler(i,j), i=0..N)],\n               j=0..N)],axes=BOXED
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'fehlerGR6$%\"iG%\"jG6\"6$%)op
eratorG%&arrowGF)7%&%\"xG6#9$&%\"yG6#9%,&-%%subsG6$%$lsgG&%\"uG6$F1F5
\"\"\"-%%u_anG6$F.F2!\"\"F)F)F)" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 
300 {PLOTDATA 3 "6$-%%MESHG6#7'7'7%$\"\"!F*F)F)7%$\"+++++D!#5F)F)7%$\"
+++++]F.F)F)7%$\"+++++vF.F)F)7%$\"\"\"F*F)F)7'7%F)F,F)7%F,F,$\"*&H&f;&
!#67%F0F,$\"*vgdI(F=7%F3F,F;7%F6F,F)7'7%F)F0F)7%F,F0$\")`I%)**F.7%F0F0
$\"**R*>T\"F.7%F3F0FF7%F6F0F)7'7%F)F3F)7%F,F3$\"*(RNr6F.7%F0F3$\"*nWll
\"F.7%F3F3FP7%F6F3F)7'7%F)F6F)7%F,F6F)7%F0F6F)7%F3F6F)7%F6F6F)-%*AXESS
TYLEG6#%$BOXG" 1 2 0 1 10 0 2 1 1 2 2 1.000000 45.000000 45.000000 0 
0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}}
{MARK "2 9 8 0 0" 0 }{VIEWOPTS 1 1 0 2 1 1805 1 1 1 1 }{PAGENUMBERS 0 
1 2 33 1 1 }