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Boundary Singularities in Linear Elliptic Differential Equations

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Boundary Singularities in Linear Elliptic Differential Equations
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  /.  Inst. Maths Applies  (1969) 5, 340-350 Boundary Singularities in Linear Elliptic Differential Equations L. Fox AND R. SANKAR Oxford University Computing Laboratory, Parks  Road,  Oxford [Received 25 March 1968]Finite-difference methods are relatively inefficient in the neighbourhood of boundarysingularities in elliptic problems. A combination of special treatment near the singularity,based on local satisfaction of the differential equation and boundary conditions, is herematched with finite-difference formulae in the rest of the field. The method is appliedto a general self-adjoint equation with either Dirichlet, Neumann or mixed conditionson parts of the boundary consisting of two straight lines meeting at the singular point.A practical problem, formerly solved by more extensive labour, illustrates the power ofthe method.1. Introduction IT  IS WELL-KNOWN  that in many elliptic problems the presence of some forms ofboundary discontinuities or singularities may not be serious, in the sense that thetrue solution is perfectly well-behaved at any interior point of the bounded region.Even with the use of numerical methods, of the finite-difference type, the inaccuraciesof the computed solution due to the presence of the singularity are then usuallysignificant only in a certain region of infection , and at sufficient distances fromthe offending point our computed results are reasonably satisfactory.On the other hand there is no doubt that in such circumstances we have threesignificant problems. First, for success we need a small finite-difference interval, atleast in the region of infection. Second, we can never by this method get very accurateresults at points in the neighbourhood of the singularity. Third, our error analysis,which is based on estimates of some derivatives of the true solution, breaks down orbecomes considerably more difficult if these derivatives become infinite at a point onthe boundary.It appears that these problems are minimized if we use finite-difference methodsonly in regions in which the solution is sufficiently well behaved in a numerical sense,and combine this with special treatment in the neighbourhood of the difficult point.This special treatment effectively determines the nature of a function which satisfiesthe differential equation and boundary conditions in the neighbourhood of thesingular point, and finds any arbitrary constants involved by matching with thefinite-difference solution.Such methods have previously been suggested for Laplace's equation in twodimensions, for example by Motz (1946), Woods (1953), Wasow (1957) and Volkov(1963). Here we extend the theory and its application to the treatment of a moregeneral self-adjoint elliptic problem in two dimensions, for which some part of the 340   a t   U ni  v  er  s i   t   y  of   S  o u t  h  C  ar  ol  i  n a- C  ol   um b i   a onN  ov  em b  er 2 4  ,2  0 1  0 i  m am a t  . ox f   or  d  j   o ur n al   s . or  gD  ownl   o a d  e d f  r  om   BOUNDARY SINGULARITIES IN ELLIPTIC PROBLEMS 341 boundary consists of two straight lines, intersecting at a singular point. We considerthe effect of various forms of boundary conditions on these two lines.To illustrate our suggestions we find the complete solution for the flow of in-compressible fluid past a screw propellor, a problem formerly solved by more extensivelabour by Goldstein (1929) and Wijngaarden (1956). 2.  Solution in the Neighbourhood of a SingularityWe treat the self-adjoint equation V 2 u  =  -g(r,9)u,  (1)where g(r,ff)=  £  g n {ey,  (2) n-0 -  d 2 u Idu  1  d 2 u Wu  =  6? + -r8-r  +  ?W  (3) and the srcin of polar co-ordinates is taken at the intersection of the lines 0 = 0, 6\=  co,  which form part of the boundary. We seek the solution of (1) subject to threedifferent sets of boundary conditions on these two lines, given respectively by(i)  u = F(r)  on  9 =  0, u =  H(r)  on  9  = co, (4)  u lP e  =  F W  oa e =   > ;Po =  H W  oa e =  } >   13M (iii)  u = F(r)  on 0 = 0, -— =  H(r)  on  9 =  co.  (6) r  o It is assumed that the functions of  r  in (4)-(6) have the convergent expansions F(r)  =  t  fnr * ,  H(r)  = £  h n ^\  j8,y > 0, (7) and we seek a solution in one of the forms «=  1  KtfV* 1 ,  u = t  r° + J{(logr)A XiJ (9)+B ai j(9)},  (8) which appear to be sufficient to cover all possibilities.Substitution of the first of (8) into (1) gives the equation U + 2]r° +m  = - £ E  0-AJK* .  (9) m = 0  IB-0  V/ = O / where the primes denote differentiation with respect to  9.  Then if the first of (8) is asolution of (1) we must have , = 0 £  9 m -iK}>  m  = 0,1,2,... y-o (10)   a t   U ni  v  er  s i   t   y  of   S  o u t  h  C  ar  ol  i  n a- C  ol   um b i   a onN  ov  em b  er 2 4  ,2  0 1  0 i  m am a t  . ox f   or  d  j   o ur n al   s . or  gD  ownl   o a d  e d f  r  om   342  L. FOX AND R.  SANKAR The same treatment with the second of (8) shows that  A atj (9)  should still satisfyequations (10), while the equations for the  B at fQ)  coefficients are given byã (11)We proceed to develop solutions of the relevant equations (10) and (11), for usein the neighbourhood of the point of intersection of the boundary lines 0 = 0 and 9  =  (o,  for each set of conditions (i) (Dirichlet conditions), (ii) (Neumann conditions)and (iii) (mixed conditions) given in equations (4), (5) and (6) respectively. Case  (0The simplest solution of (10), given by the first of (8), which satisfies the simpleconditions«=//>+/» on  9 =  0, u = 0 on  0 =  <o,  (12)is clearly obtained by takinga =  /;+/?,  Ao(O)=/ B)  A., 0 (fo) = 0 A a ,0)  = 0,  A^ico)  = 0,  j=  1,2,... For then, in the first of  (8),  the term ^ I .o(0>' I+p  satisfies (12), and the other terms inthe series vanish on  9  = 0 and  9  =  a>.  From the first of  (10)  we then find A (0) 4,oW - /.  sin(fI+/Oa)  ' and we can easily solve for the other  Aaj(ff),j >  0, to satisfy the rest of (10) and thesecond of  (13).  We denote the resulting solution bywhere the symbol  D  refers to the Dirichlet case.The possible vanishing of  the  denominator in (15) illustrates the need for a solutionof the type of the second of (8). Suppressing the details, we find that the solutionreplacing (15) is then given by  (r ' 0)  a>cos(n+p)co x D[r + {Gog r) sin  ( B  + j8)(0-<»)+(0-a>) cos (n+0)(0-a))}]. (16)The corresponding solutions for  the  boundary conditions u  = 0 on 0 = 0, u = /V n+T  on 0 = co (17)are given by(18)r) sin («+y)0+0 cos (n+y)0}]. (19)   a t   U ni  v  er  s i   t   y  of   S  o u t  h  C  ar  ol  i  n a- C  ol   um b i   a onN  ov  em b  er 2 4  ,2  0 1  0 i  m am a t  . ox f   or  d  j   o ur n al   s . or  gD  ownl   o a d  e d f  r  om   BOUNDARY SINGULARITIES IN ELLIPTIC PROBLEMS 343 We must also include solutions which vanish on both 0 = 0 and  6  =  a>.  For thesewe clearly take a  = — ,  A a  0  = sin — 0, W CO where  m  is  an integer, and the remaining  Aa,j,  for./  > 0, follow as before.The complete solution of case (i) is then given by u(r,9)  = £'  W sin  (-A],  20) (21)where the c ra  are arbitrary constants, and where the prime denotes that  ui l)  or  u{ 2)  isreplaced by  u^  or i7< 2)  when necessary. Clearly, for similar requirements for boundedsolutions, the integer  m  cannot be negative. Cases  (ii) and  (Hi) For case (ii) we replace the symbol  D  by the symbol  N  (for Neumann), and withsimilar analysis and notation we find the solution <r,ff)  =  £'  W\r,9)+ui 2 \r,9)}+  Z n = 0 m-0 wherecos  22) sin  {n co U  ( i  cos  (n+p+l)o) x N[r + + '{(log r) cos (n+^ +1X0-Q3)-(0-co) sin  (n+)3  +1X0-©)} , 23) and o  L -K COS  (n+y+l)co . 24) For case (iii) the relevant symbol is  M,  representing Mixed conditions, and wefind the solution u(r,9)  = £'  { u n 1 Xr,9)  +  u in2 Xr,9)}+  £ cj r 0  m—0 where(25) CD  sin  (n + P)oo x M[r + '{(log r)  cos  (n+pX0-co)-(0-o) sin (26)   a t   U ni  v  er  s i   t   y  of   S  o u t  h  C  ar  ol  i  n a- C  ol   um b i   a onN  ov  em b  er 2 4  ,2  0 1  0 i  m am a t  . ox f   or  d  j   o ur n al   s . or  gD  ownl   o a d  e d f  r  om 
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