-- FreeFem++ v4.14 (mer. 06 mars 2024 16:59:04 CET - git v4.14-1-g2b2052ae)
   file : mitchell_07.edp
 Load: lg_fem lg_mesh lg_mesh3 eigenvalue 
    1 : //  mitchell_07.edp
    2 : //
    3 : //  Discussion:
    4 : //
    5 : //    -uxx - uyy = f on the unit square.
    6 : //    u  = g on the boundary.
    7 : //
    8 : //    f = - alpha * ( alpha - 1 ) * x^(alpha-2)
    9 : //    g = x^alpha
   10 : //
   11 : //    The parameter alpha should be 0.5 or greater.  It determines the
   12 : //    strength of the singularity.  The suggested value is alpha = 0.6
   13 : //
   14 : //  Licensing:
   15 : //
   16 : //    This code is distributed under the MIT license.
   17 : //
   18 : //  Modified:
   19 : //
   20 : //    19 December 2014
   21 : //
   22 : //  Author:
   23 : //
   24 : //    John Burkardt
   25 : //
   26 : //  Reference:
   27 : //
   28 : //    Frederic Hecht,
   29 : //    Freefem++,
   30 : //    Third Edition, version 3.22
   31 : //
   32 : //    William Mitchell,
   33 : //    A collection of 2D elliptic problems for testing adaptive
   34 : //    grid refinement algorithms,
   35 : //    Applied Mathematics and Computation,
   36 : //    Volume 220, 1 September 2013, pages 350-364.
   37 : //
   38 : cout << "\n";
   39 : cout << "mitchell_07():\n";
   40 : cout << "  FreeFem++ version\n";
   41 : cout << "  Solve the boundary line singularity problem.\n";
   42 : 
   43 : border bottom ( t = 0.0, 1.0 )   { x = t;    y = 0.0; label = 1; }
   44 : border right  ( t = 0.0, 1.0 )   { x = 1.0;  y = t;   label = 1; }
   45 : border top    ( t = 1.0, 0.0 )   { x = t;    y = 1.0; label = 1; }
   46 : border left   ( t = 1.0, 0.0 )   { x = 0.0;  y = t;   label = 1; }
   47 : //
   48 : //  Define Th, the triangulation of the "left" side of the boundaries.
   49 : //
   50 : int n = 10;
   51 : mesh Th = buildmesh ( bottom ( n ) + right ( n ) + top ( n ) + left ( n ) );
   52 : //
   53 : //  Define Vh, the finite element space defined over Th, using P1 basis functions.
   54 : //
   55 : fespace Vh ( Th, P1 );
   56 : //
   57 : //  Define U, V, and F, piecewise continuous functions over Th.
   58 : //
   59 : Vh u;
   60 : Vh v;
   61 : //
   62 : //  Define parameter.
   63 : //
   64 : real alpha = 0.6;
   65 : //
   66 : //  Define function F.
   67 : //
   68 : func f = - alpha * ( alpha - 1.0 ) * pow ( x, alpha - 2 );
   69 : //
   70 : //  Define function G.
   71 : //
   72 : func g = pow ( x, alpha );
   73 : //
   74 : //  Solve the variational problem.
   75 : //
   76 : solve Laplace ( u, v ) 
   77 :   = int2d ( Th ) ( dx(u)*dx(v) + dy(u)*dy(v) )
   78 :   - int2d ( Th ) ( f * v ) 
   79 :   + on ( 1, u = g );
   80 : //
   81 : //  Plot the solution.
   82 : //
   83 : plot ( u, wait = true, fill = true, ps = "mitchell_07_u.ps" );
   84 : //
   85 : //  Plot the mesh.
   86 : //
   87 : plot ( Th, wait = true, ps = "mitchell_07_mesh.ps" );
   88 : //
   89 : //  Save the mesh file.
   90 : //
   91 : savemesh ( Th, "mitchell_07.msh" );
   92 : //
   93 : //  Terminate.
   94 : //
   95 : cout << "\n";
   96 : cout << "mitchell_07():\n";
   97 : cout << "  Normal end of execution.\n";
   98 : 
   99 : exit ( 0 );
  100 :  sizestack + 1024 =2616  ( 1592 )


mitchell_07():
  FreeFem++ version
  Solve the boundary line singularity problem.
  --  mesh:  Nb of Triangles =    240, Nb of Vertices 141
  -- Solve : 
          min 1.25594e-31  max 1
  number of required edges : 0

mitchell_07():
  Normal end of execution.
  current line = 99
exit(0)
 err code 0 ,  mpirank 0
 CodeAlloc : nb ptr  3987,  size :525840 mpirank: 0
Ok: Normal End