-- FreeFem++ v4.14 (mer. 06 mars 2024 16:59:04 CET - git v4.14-1-g2b2052ae)
   file : mitchell_02c.edp
 Load: lg_fem lg_mesh lg_mesh3 eigenvalue 
    1 : //  mitchell_02c.edp
    2 : //
    3 : //  Discussion:
    4 : //
    5 : //    The Laplacian operator is applied on a square with a reentrant corner.
    6 : //
    7 : //    The geometry is defined by an internal angle PI < OMEGA <= 2PI.
    8 : //    
    9 : //    -uxx - uyy = f in the region.
   10 : //    u = g on the boundary.
   11 : //
   12 : //    F(X,Y) = 0
   13 : //    ALPHA = PI / OMEGA
   14 : //    R = sqrt ( X^2 + Y^2 )
   15 : //    THETA = arctan ( Y / X )
   16 : //    G(X,Y) = R^ALPHA * SIN ( ALPHA * THETA)
   17 : //
   18 : //  Licensing:
   19 : //
   20 : //    This code is distributed under the MIT license.
   21 : //
   22 : //  Modified:
   23 : //
   24 : //    24 May 2020
   25 : //
   26 : //  Author:
   27 : //
   28 : //    John Burkardt
   29 : //
   30 : //  Reference:
   31 : //
   32 : //    Frederic Hecht,
   33 : //    Freefem++,
   34 : //    Third Edition, version 3.22
   35 : //
   36 : //    William Mitchell,
   37 : //    A collection of 2D elliptic problems for testing adaptive
   38 : //    grid refinement algorithms,
   39 : //    Applied Mathematics and Computation,
   40 : //    Volume 220, 1 September 2013, pages 350-364.
   41 : //
   42 : cout << "\n";
   43 : cout << "mitchell_02c():\n";
   44 : cout << "  FreeFem++ version\n";
   45 : cout << "  Reentrant corner\n";
   46 : 
   47 : real omega = 6.0 * pi / 4.0;
   48 : 
   49 : cout << "  Omega = " << omega << "\n";
   50 : int p = 5;
   51 : 
   52 : border b1 ( t = 0.0, +1.0 )   { x = t;    y = 0.0; label = 1; }
   53 : real l1 = 1.0;
   54 : int n1 = ( round ) ( l1 * p + 1 );
   55 : cout << "  N1 = " << n1 << "\n";
   56 : border b2 ( t = 0.0, +1.0 )   { x = +1.0; y = t;   label = 1; }
   57 : real l2 = 1.0;
   58 : int n2 = ( round ) ( l2 * p + 1 );
   59 : cout << "  N2 = " << n2 << "\n";
   60 : border b3 ( t = 1.0, -1.0 )   { x = t;    y = 1.0; label = 1; }
   61 : real l3 = 2.0;
   62 : int n3 = ( round ) ( l3 * p + 1 );
   63 : cout << "  N3 = " << n3 << "\n";
   64 : //
   65 : //  What we do depends on OMEGA!
   66 : //
   67 :   real xc = - cos ( omega ) / sin ( omega );
   68 :   real yc = -1.0;
   69 :   real rc = -1.0 / sin ( omega );
   70 : 
   71 :   border b4 ( t = 1.0, -1.0 ) { x = -1.0; y = t;   label = 1; }
   72 :   real l4 = 2.0;
   73 :   int n4 = ( round ) ( l4 * p + 1 );
   74 :   border b5 ( t = -1.0, xc  ) { x = t; y = yc; label = 1; }
   75 :   real l5 = xc + 1.0;
   76 :   int n5 = ( round ) ( l5 * p + 1 );
   77 :   border b7 ( t = rc, 0.0 ) { x = t * cos ( omega ); y = t * sin ( omega ); label = 1; }
   78 :   real l7 = rc;
   79 :   int n7 = ( round ) ( l7 * p + 1 );
   80 : 
   81 :   mesh Th = buildmesh ( b1 ( n1 ) + b2 ( n2 ) + b3 ( n3 ) + b4 ( n4 ) 
   82 :                       + b5 ( n5 ) + b7 ( n7 ) ); 
   83 : //
   84 : //  Define Vh, the finite element space defined over Th, using P1 basis functions.
   85 : //
   86 : fespace Vh ( Th, P1 );
   87 : //
   88 : //  Define U, V, and F, piecewise continuous functions over Th.
   89 : //
   90 : Vh u;
   91 : Vh v;
   92 : //
   93 : //  Define the right hand side function F.
   94 : //
   95 : func f = 0.0;
   96 : //
   97 : //  Define the boundary condition function G.
   98 : //
   99 : real alpha = pi / omega;
  100 : func g = pow ( x * x + y * y, alpha / 2.0 ) * sin ( alpha * atan2 ( y, x ) );
  101 : //
  102 : //  Solve the variational problem.
  103 : //
  104 : solve Laplace ( u, v ) 
  105 :   = int2d ( Th ) ( dx(u)*dx(v) + dy(u)*dy(v) )
  106 :   - int2d ( Th ) ( f * v ) 
  107 :   + on ( 1, u = g );
  108 : //
  109 : //  Plot the solution.
  110 : //
  111 : plot ( u, wait = true, fill = true, ps = "mitchell_02c_u.eps" );
  112 : //
  113 : //  Plot the mesh.
  114 : //
  115 : plot ( Th, wait = true, ps = "mitchell_02c_mesh.eps" );
  116 : //
  117 : //  Save the mesh file.
  118 : //
  119 : savemesh ( Th, "mitchell_02c.msh" );
  120 : //
  121 : //  Terminate.
  122 : //
  123 : cout << "\n";
  124 : cout << "mitchell_02c():\n";
  125 : cout << "  Normal end of execution.\n";
  126 : 
  127 : exit ( 0 );
  128 :  sizestack + 1024 =1976  ( 952 )


mitchell_02c():
  FreeFem++ version
  Reentrant corner
  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

mitchell_02c():
  Normal end of execution.
  current line = 127
exit(0)
 err code 0 ,  mpirank 0
 CodeAlloc : nb ptr  4176,  size :531232 mpirank: 0
Ok: Normal End