Appled Mathematcal Scences Vol. 6 01 no. 8 369 380 New Method for Solvng Posson Equaton on Irregular Domans J. Izadan and N. Karamooz Department of Mathematcs Facult of Scences Mashhad BranchIslamc Azad Unverst Mashhad Iran M. Jall Department of Mathematcs Neshabur Branch Islamc Azad Unverst Neshabur Iran Jall.maram@ahoo.com Abstract In ths paper a new method for solvng Posson equaton wth Drchlet condtons on non-rectangular domans s presented. For ths purpose two numercal dfferentaton methods are ntroduced for non-equdstant ponts. These two tpes of numercal dfferentatons are used for solvng Posson problems on rregular domans wth two tpes of meshes : rregular and sem-rregular. In ths paper the numercal dfferentaton method applng non-equdstant ponts are ntroduced. The numercal results show the effcenc and performance of proposed method. Kewords: Posson Equaton Drchlet condton Fnte dfferences method (FDM) Irregular domans 1 Introducton One of the most mportant partal dfferental equatons s Posson equaton that has great applcatons n scence. There s a varet of papers related to ths subject. Ths equaton belong to ellptc PDE s group. There are varous numercal methods for solvng these equatons FEM BEM etc. For a fundamental revew of these methods one can refer to Ames [1] McDonough [3].
370 J. Izadan and N. Karamooz and M. Jall Applng FEM s sutable for rregular domans but It does use a lot of tme consumng and creatng the mesh ponts s a task that s reserved to specalsts. Usng fnte dfferences method (FDM) s possble for rectangular domans. However we can use fnte dfferences for rregular domans b usng sub-regons methods [34]. In ths paper we develop a partcular tpe of fnte dfference methods adequate for solvng ellptc problems on rregular domans. For ths purpose certan numercal dfferentaton rules s ntroduced b usng nterpolaton of functons on an rregular fve ponts mesh (or molecule).we organze the present paper as follows. After ntroductor secton n secton two rregular mesh dfferentaton rules wll be ntroduced. In secton 3 two numercal methods wll be presented for solvng Posson equaton wth Drchlet boundar condtons. In secton 4 the numercal experments are presented for three specal domans and fnall secton 5 s concluson and dscusson the paper. Numercal dfferentaton formulae A great categor of fnte dfferences methods are based on three ponts fnte dfference dervatves that for a functon u : R R can be gven b : 1 ux ( x ) = u( x + h ) u( x h ) + O ( h ) h 1 u ( x ) = u( x + k ) u( x k ) + O( k ) k 1 uxx ( x ) = u ( x + h ) + u( x h ) u( x ) + O ( h ) h 1 u ( x ) = u ( x + k ) + u( x k ) u( x ) + O( k ) k where handk are step length of ponts n x axs and axs respectvel. Hence we can wrte the dscretzed fnte dfference equaton on a rectangular doman for Posson equaton u f Δ = as followng : ( ) k u x+ h + u x h + h u x + h + u x k h + k u x = h k f x Now we consder a star mesh that contans fve ponts (fg.1) ( x j )( x 1 j )( x + 1 j )( x j+ 1) ( x j 1) =01 n j=1 m Ths procedure fals when the doman s not rectangular shape. Therefore we need the partcular numercal dervatves n ths case :
New method for solvng Posson equaton 371 Fg.1 5 ponts molecule The non-equal step length of ponts n x axs and axs are gven as follows respectvel : h1 = x x 1 h = x+ 1 x k = k =. 1 j j 1 j+ 1 j We desre to compute approxmaton of ux ( x j ) and ( j ) error O ( H ) such that H = max h h k k 1 1 u x wth an B applng at most three functonal values on the molecule as follows : (11) (1) () ( ) = α ( 1 ) + α ( ) + α ( + 1 ) + (1) () (3) = α ( 1) + α + α ( + 1) + u x u x u x u x O H x j j j j j j j u x u x u x u x O H. (1) j j j j j j j Usng Talor expanson u n ( j ) x and solvng lnear sstems the unknown coeffcents are determned wth followng expressons : h h h h = ( αj αj αj ) (11) (1) () 1 1 h1 + h1h h1h h1h + h k k k k = ( αj αj αj ) (1) () (3) 1 1 k1 + k1k k1k k1k + k
37 J. Izadan and N. Karamooz and M. Jall Smlar arguments can be used for computng the approxmate values of u x. Acceptng uxx ( x j ) and ( j ) (11) (1) () ( ) = β ( 1 ) + β ( ) + β ( + 1 ) + (1) () (3) = β ( 1) + β + β ( + 1) + u x u x u x u x O H xx j j j j j j j u x u x u x u x O H () j j j j j j j We can fnd out easl that : = h1 + h1h h1h h1h + h =. k1 + k1k k1k k1k + k (11) (1) () ( βj βj βj ) (1) () (3) ( βj βj βj ) We obtaned for rregular meshes from a ordnar fnte dfference dscretzaton a b-dmensonal doman. The numercal partal dervatves can be calculated as mentoned above wth a smple nterpolaton technque that can be called nonequdstant fnte dfferences. But for a general meshes wth molecules lke fgure. Computng numercal dervatves are more complcated. For ths purpose we consder 5 dstnct ponts ( x )( x )( x )( x )( x ) + 1 + 1 + + + 3 + 3 + 4 + 4 These ponts are not stuated on a sngle lne (fg..). Fg. rregular 5 ponts molecule To compute approxmatons of smple partal dervatves u x u u xx and u the nterpolaton concept are used for fve gven ponts. These approxmatons are as follow :
New method for solvng Posson equaton 373 ( l k ) ( l k ) = + + ( l k ) ( l k ) + α u( x ) + α u( x ) ( l k ) D x α u x α u x α u x lk 1 + 1 + 1 3 + + (3) 4 + 3 + 3 5 + 4 + 4 where lk ( ) l wth respect to k drecton at ponts ( ) center ( ) D x are approxmate values of smple partal dervatve of order x. Expandng Talor seres u wth x and equal to rght hand sdes of equaton (3) wth the correspondng partal dervatve certan terms and wth truncaton error of m O ( H ) m = 1 for l = and m = for l = 1 we obtan a sstems of fve k l lnear equatons wth fve unknowns α j j = 1 K 5 whch can be solved b Cramer rule. The obtaned values are the followng matrx equaton solutons : α ( l k ) 1 1 1 1 1 1 ( l k ) 0 h h3 h4 h5 α ( l k ) 0 k k k k α = b ( l k 0 h ) h3 h4 h 5 α4 ( l k ) ( l k ) 3 4 5 3 0 k k k k α 3 4 5 5 lk where b s a vector gven as : b b b b ( 11) ( 1) ( 1) ( 11) = = = = ( 01000) ( 00100) ( 000 0) t ( 0000 ) One can prove easl two followng proposton about truncaton error of two proposed numercal dfferentaton methods..1 Proposton : If f C 3 [ a b] and and a x1 x x 3 b then ( α 11 α 11 α 11 ) ( ) f x f x + f x + f x = O H 1 1 3 3 t t t
374 J. Izadan and N. Karamooz and M. Jall ( 1) ( 1) ( 1) ( α α α ) f x f x + f x + f x = O H 1 1 3 3 H = max h h h = x x h = x x 1 1 1 3 The above proposton s related to the functons of a varable t can be generalzed to n -varable functons. { }. Proposton : If ( ) ( ) ( ) ( ) B x = x R x x < ε and 1 1 1 1 and D ( x ) s defned as (3) and ( x ) f C B then lk 1 1 ( l k ) ( ) ( ) = u x D x O H 1 1 l k 1 1 B = 1 3 45 and H = max h h h h k k k k 1 3 4 1 3 4 where x u x = u x u x = u x 11 1 1 1 1 1 1 1 1 1 u x = u x u x = u x 1 1 1 xx 1 1 1 1 1 1 h = x x k = = 4. + 1 1 + 1 1 3 Non-equdstant fnte dfferences method We consder the followng problem : D s a regon n Δ u x = f x x D ux = g x x D R that can to be multple-connected. f C 0 ( D) g C 0 ( D) % where D % s a regon contanng D f and g are gven functons. When D s a rectangular doman [ ] [ ] D = a b c d
New method for solvng Posson equaton 375 The numercal fnte dfference soluton s trval. We suppose D s not a rectangular doman but a doman that embedded n the doman : ~ [ ] [ ] D = ab cd more precsel we accept that projecton of D over x and axs are [ ab ] and [ cd ] respectvel. For sake of smplct such non rectangular doman D s called rregular. Now two tpes of meshes can be recognzed. The frst tpe s the mesh that s nduced b dscretzaton of ~ D as we consder for ordnar fnte dfferences method. These meshes wll be called sem-rregular meshes. The next tpe s the mesh wth arbtrar mesh ponts ( see fg3.1 and 3.). Ths mesh wll be called rregular mesh.for ever tpe one apples ts approprate approach. Fg 3.1 Sem-rregular mesh Fg 3. rregular mesh 3.1 Sem-rregular mesh approach Havng consdered rregular doman D and rectangular doman D ~ that have been ab ntroduced n prevous paragraphs we consder two unform parttons of [ ] and [ cd ] are ntroduced : a = x < x < K< x = b c = < < K < = d 0 1 n 0 1 wth step lengths h and k respectvel. Two tpes of ponts for D are to be recognzed b constructng a rectangular mesh on ~ D we observe two groups of ponts for D : nternal ponts and external ponts. Dscardng the external ponts and consderng the ponts of ntersecton of lnes. m
376 J. Izadan and N. Karamooz and M. Jall x = x = j = 01 K n j = 1 K m wth boundar of D. Ths set of ponts consttutes the desred mesh ponts for our approach. Ths mesh s called sem-rregular mesh. Ths mesh can be denoted b D hk consstng of nternal ponts that s denoted b D o hk and the set of boundar ponts that are denoted b D hk Now wrtng Posson equaton and ts boundar condtons on the mesh ponts one fnds u x + u x = f x x D (4) o xx h k u x = g x x D (5) o h k If N s the number of mesh ponts D hk then b substtutng the numercal dervatves N equatons wth N unknowns are found. The unknowns are values of u on mesh ponts. Here we consder two tpes of nternal mesh ponts: the ponts that are just next to boundar and the rest. For frst group of ponts non-equdstant three ponts numercal dfferentaton rules gven n secton. and for second group of ponts the ordnar central fnte dfference numercal dfferentaton rules are appled. B substtutng proper numercal dervatves n (4) one consttutes a sstems of N equatons and N unknowns. B solvng ths sstems wth ts proper methods the desred approxmated values of u on mesh ponts can be determned. An exact orderng of mesh ponts s essental. Numberng of mesh ponts can be done n horzontal levels of mesh ponts from left to rght and from down to up. Three numercal experments for crcular regon and annular regon are presented n secton 4. It s worth to menton that n ths approach the results are ver sensble to boundar ponts of mesh. One must be ver careful to consder the proper star molecule for ponts n vcnt of boundar. 3. Irregular mesh approach In ths approach one has the ablt to choose the mesh ponts. Thus constructng a mesh can be dctated b dfferent exgences for example the necesst of havng the value of soluton n partcular ponts or the lmtatons mposed b geometr of gven doman. But ths freedom can be lmted b bad condtonng of small matrces that are used for calculatng 5 ponts numercal dervatves that have been ntroduced n secton. For successful mplementaton of ths method t s mportant to create an algorthm that can furnsh us four neghbors of the gven pont( x ). Then for ever set of such 5 ponts molecule we need a local numercal numberng.
New method for solvng Posson equaton 377 Now for consttutng the equatons of desred lnear sstem for a gven 1... N x s an nternal ponts of mesh after determnng four = f ( ) neghborng ponts of ( x ) as ( 1) ( ) ( 1) ( ) ( α1 + α1 ) u( x ) + ( α + α ) u( x + 1 + 1 ) ( 1) ( ) ( 1) ( ) + ( α + α ) + α + α x + l + l l = 4 then (4) elds : u x + + ( ) u( x + + ) ( 1) ( ) ( α ) 5 α 5 + 4 + 4 3 3 4 4 3 3 + + u x = f x = 1 K n (6) If ( ) x s a boundar ponts the boundar condtons are wrtten as : u x = g x = 1 K m (7) where n s the number of nternal mesh ponts and m s the number of boundar mesh ponts. If N = n+ m then N equatons (6) and (7) gve a sstem of N lnear equatons wth N unknowns that can be solved b one of drect or teratve methods. 4 Numercal experments In ths secton we present three examples on two dfferent domans : a smpl connected and multple-connected doman precsel a crcular doman and an * annular doman. u u are approxmate soluton and exact soluton n ( x ) respectvel. 4.1 Example We consder the followng problem xx u + u = 4 x D u x = x + x D {( ) 0.5} D = x R x + < Ths problem s solved b frst approach the numercal results and errors are gven n the table 4.1. The exact soluton s u( x ) = x +. The mesh grds of D and plot of surface of soluton s presented n fgure 4.1 and 4. respectvel. The maxmum absolute value of errors s and CPU- tme s 11 seconds.
378 J. Izadan and N. Karamooz and M. Jall Table 4.1 Numercal result of example 4.1 n some ponts. n ( x ) u u 1 (-0.-0.5) 0.0176 (-0.46-0.) 0.183 3 (-0.1-0.1) 0.0649 4 (0.50) 0.0036 5 (-0.10.) 0.3059 6 (-0.460.) 0.035 7 (0.0.5) 0 * Fg 4.1 mesh ponts of D of example 4.1 nternal and boundar ponts Are presented wth + and * respectvel. Fgure 4. surface of soluton of example 4.1
New method for solvng Posson equaton 379 4. Example One choose the problem 4.1 wth D defned as follows : {( ) 1 4} D = x R < x + < The numercal results are gven wth table 4.. Table 4. Numercal result of example 4. n ( x ) u u 1 (0.8-1) 0.0010 (1-1.6) 0.0044 3 (-1.4) 0.0060 4 (-1.59-1) 0.0038 5 (0.-1.) 0.0009 6 (-1.84-0.8) 0.0085 7 (0.41.94) 0 * Ths example are executed wth two gven approachs. The meshgrds of D and plot of surface soluton s presented n fg 4.3 and 4.4 respectvel. The maxmum absolute value of errors s and CPU-tme s 44.37 seconds. Fg 4.3 Mesh ponts of D of example 4. nternal and boundar ponts Are presented wth + and * respectvel.
380 J. Izadan and N. Karamooz and M. Jall Fg 4.4 Surface of soluton of example 4. 5 Concluson and dscusson In ths paper two new rules for numercal dfferentaton are ntroduced. The both are appled for fndng numercal soluton of Posson equaton wth Drchlet condtons. Numercal experments show a good performance and effcenc of these two methods. However the frst approach s less complcated but more sensble to geometr of boundar and the boundar mesh ponts. The second approach s much easer to appl geometrcall but s more complcated computatonall. However these methods can be used for solvng PDE s on nonrectangular domans. References [1] Ames W.F. Numercal Methods For Partal Dfferental Equaton Academc Press INC.1997. [] Hman J.M. and LarrouturouB.The Numercal Dfferentaton of Dscrete Functon Usng Polnomcal Interpolaton Methods Journal Of Appled Mathematcs and Computaton New York198. [3] McDonough J.M. Lectures on Computatonal Numercal Analss of partal Dfferental Equatons Unverst of Kentuck 008. [4] Poplau G Multgrd Posson solver for Non-Equdstant Tensor Product Meshes.November 003. Receved: June 011