-- FreeFem++ v  3.320001 (date Ven  7 nov 2014 14:28:55 CET)
 Load: lg_fem lg_mesh lg_mesh3 eigenvalue 
    1 : //  Discussion:
    2 : //
    3 : //    The Laplacian operator is applied on a square with a reentrant corner.
    4 : //
    5 : //    The geometry is defined by an internal angle PI < OMEGA <= 2PI.
    6 : //    
    7 : //    ALPHA = PI / OMEGA
    8 : //    R = sqrt ( X^2 + Y^2 )
    9 : //    THETA = arctan ( Y / X )
   10 : //    U(X,Y) = R^ALPHA * SIN ( ALPHA * THETA)
   11 : //
   12 : //    Dirichlet boundary conditions.
   13 : //
   14 : //    Surprisingly, U(X,Y) satisfies the equation:
   15 : //
   16 : //      - Uxx - Uyy = 0
   17 : //
   18 : //  Location:
   19 : //
   20 : //    http://people.sc.fsu.edu/~jburkardt/examples/mitchell_freefem++/test02c.edp
   21 : //
   22 : //  Licensing:
   23 : //
   24 : //    This code is distributed under the GNU LGPL license.
   25 : //
   26 : //  Modified:
   27 : //
   28 : //    17 December 2014
   29 : //
   30 : //  Author:
   31 : //
   32 : //    John Burkardt
   33 : //
   34 : //  Reference:
   35 : //
   36 : //    Frederic Hecht,
   37 : //    Freefem++,
   38 : //    Third Edition, version 3.22
   39 : //
   40 : //    William Mitchell,
   41 : //    A collection of 2D elliptic problems for testing adaptive
   42 : //    grid refinement algorithms,
   43 : //    Applied Mathematics and Computation,
   44 : //    Volume 220, 1 September 2013, pages 350-364.
   45 : //
   46 : real omega = 6.0 * pi / 4.0;
   47 : 
   48 : cout << "  Omega = " << omega << "\n";
   49 : int p = 5;
   50 : 
   51 : border b1 ( t = 0.0, +1.0 )   { x = t;    y = 0.0; label = 1; }
   52 : real l1 = 1.0;
   53 : int n1 = ( round ) ( l1 * p + 1 );
   54 : cout << "  N1 = " << n1 << "\n";
   55 : border b2 ( t = 0.0, +1.0 )   { x = +1.0; y = t;   label = 1; }
   56 : real l2 = 1.0;
   57 : int n2 = ( round ) ( l2 * p + 1 );
   58 : cout << "  N2 = " << n2 << "\n";
   59 : border b3 ( t = 1.0, -1.0 )   { x = t;    y = 1.0; label = 1; }
   60 : real l3 = 2.0;
   61 : int n3 = ( round ) ( l3 * p + 1 );
   62 : cout << "  N3 = " << n3 << "\n";
   63 : //
   64 : //  What we do depends on OMEGA!
   65 : //
   66 :   real xc = - cos ( omega ) / sin ( omega );
   67 :   real yc = -1.0;
   68 :   real rc = -1.0 / sin ( omega );
   69 : 
   70 :   border b4 ( t = 1.0, -1.0 ) { x = -1.0; y = t;   label = 1; }
   71 :   real l4 = 2.0;
   72 :   int n4 = ( round ) ( l4 * p + 1 );
   73 :   border b5 ( t = -1.0, xc  ) { x = t; y = yc; label = 1; }
   74 :   real l5 = xc + 1.0;
   75 :   int n5 = ( round ) ( l5 * p + 1 );
   76 :   border b7 ( t = rc, 0.0 ) { x = t * cos ( omega ); y = t * sin ( omega ); label = 1; }
   77 :   real l7 = rc;
   78 :   int n7 = ( round ) ( l7 * p + 1 );
   79 : 
   80 :   mesh Th = buildmesh ( b1 ( n1 ) + b2 ( n2 ) + b3 ( n3 ) + b4 ( n4 ) 
   81 :                       + b5 ( n5 ) + b7 ( n7 ) ); 
   82 : //
   83 : //  Define Vh, the finite element space defined over Th, using P1 basis functions.
   84 : //
   85 : fespace Vh ( Th, P1 );
   86 : //
   87 : //  Define U, V, and F, piecewise continuous functions over Th.
   88 : //
   89 : Vh u;
   90 : Vh v;
   91 : //
   92 : //  Define the right hand side function F.
   93 : //
   94 : func f = 0.0;
   95 : //
   96 : //  Define the boundary condition function G.
   97 : //
   98 : real alpha = pi / omega;
   99 : func g = pow ( x * x + y * y, alpha / 2.0 ) * sin ( alpha * atan2 ( y, x ) );
  100 : //
  101 : //  Solve the variational problem.
  102 : //
  103 : solve Laplace ( u, v ) 
  104 :   = int2d ( Th ) ( dx(u)*dx(v) + dy(u)*dy(v) )
  105 :   - int2d ( Th ) ( f * v ) 
  106 :   + on ( 1, u = g );
  107 : //
  108 : //  Plot the solution.
  109 : //
  110 : plot ( u, wait = 1, fill = true, ps = "test02c_u.eps" );
  111 : //
  112 : //  Plot the mesh.
  113 : //
  114 : plot ( Th, wait = 1, ps = "test02c_mesh.eps" );
  115 : //
  116 : //  Save the mesh file.
  117 : //
  118 : savemesh ( Th, "test02c.msh" );
  119 : 
  120 :  sizestack + 1024 =1976  ( 952 )

  Omega = 4.71239
  N1 = 6
  N2 = 6
  N3 = 11
  --  mesh:  Nb of Triangles =    224, Nb of Vertices 136
  -- Solve : 
          min -1.25992  max 1.25992
  number of required edges : 0
times: compile 0s, execution 0.01s,  mpirank:0
 CodeAlloc : nb ptr  2829,  size :347376 mpirank: 0
Ok: Normal End