-- FreeFem++ v4.6 (Thu Apr  2 15:47:38 CEST 2020 - git v4.6)
 Load: lg_fem lg_mesh lg_mesh3 eigenvalue 
    1 : //  Discussion:
    2 : //
    3 : //    -uxx - uyy = f on the unit square.
    4 : //    u  = g on the boundary.
    5 : //
    6 : //    f = - alpha * ( alpha - 1 ) * x^(alpha-2)
    7 : //    g = x^alpha
    8 : //
    9 : //    The parameter alpha should be 0.5 or greater.  It determines the
   10 : //    strength of the singularity.  The suggested value is alpha = 0.6
   11 : //
   12 : //  Location:
   13 : //
   14 : //    http://people.sc.fsu.edu/~jburkardt/freefem_src/mitchell_07/mitchell_07.edp
   15 : //
   16 : //  Licensing:
   17 : //
   18 : //    This code is distributed under the GNU LGPL license.
   19 : //
   20 : //  Modified:
   21 : //
   22 : //    19 December 2014
   23 : //
   24 : //  Author:
   25 : //
   26 : //    John Burkardt
   27 : //
   28 : //  Reference:
   29 : //
   30 : //    Frederic Hecht,
   31 : //    Freefem++,
   32 : //    Third Edition, version 3.22
   33 : //
   34 : //    William Mitchell,
   35 : //    A collection of 2D elliptic problems for testing adaptive
   36 : //    grid refinement algorithms,
   37 : //    Applied Mathematics and Computation,
   38 : //    Volume 220, 1 September 2013, pages 350-364.
   39 : //
   40 : cout << "\n";
   41 : cout << "mitchell_07\n";
   42 : cout << "  FreeFem++ version\n";
   43 : cout << "  Solve the boundary line singularity problem.\n";
   44 : 
   45 : border bottom ( t = 0.0, 1.0 )   { x = t;    y = 0.0; label = 1; }
   46 : border right  ( t = 0.0, 1.0 )   { x = 1.0;  y = t;   label = 1; }
   47 : border top    ( t = 1.0, 0.0 )   { x = t;    y = 1.0; label = 1; }
   48 : border left   ( t = 1.0, 0.0 )   { x = 0.0;  y = t;   label = 1; }
   49 : //
   50 : //  Define Th, the triangulation of the "left" side of the boundaries.
   51 : //
   52 : int n = 10;
   53 : mesh Th = buildmesh ( bottom ( n ) + right ( n ) + top ( n ) + left ( n ) );
   54 : //
   55 : //  Define Vh, the finite element space defined over Th, using P1 basis functions.
   56 : //
   57 : fespace Vh ( Th, P1 );
   58 : //
   59 : //  Define U, V, and F, piecewise continuous functions over Th.
   60 : //
   61 : Vh u;
   62 : Vh v;
   63 : //
   64 : //  Define parameter.
   65 : //
   66 : real alpha = 0.6;
   67 : //
   68 : //  Define function F.
   69 : //
   70 : func f = - alpha * ( alpha - 1.0 ) * pow ( x, alpha - 2 );
   71 : //
   72 : //  Define function G.
   73 : //
   74 : func g = pow ( x, alpha );
   75 : //
   76 : //  Solve the variational problem.
   77 : //
   78 : solve Laplace ( u, v ) 
   79 :   = int2d ( Th ) ( dx(u)*dx(v) + dy(u)*dy(v) )
   80 :   - int2d ( Th ) ( f * v ) 
   81 :   + on ( 1, u = g );
   82 : //
   83 : //  Plot the solution.
   84 : //
   85 : plot ( u, wait = 1, fill = true, ps = "mitchell_07_u.eps" );
   86 : //
   87 : //  Plot the mesh.
   88 : //
   89 : plot ( Th, wait = 1, ps = "mitchell_07_mesh.eps" );
   90 : //
   91 : //  Save the mesh file.
   92 : //
   93 : savemesh ( Th, "mitchell_07.msh" );
   94 : //
   95 : //  Terminate.
   96 : //
   97 : cout << "\n";
   98 : cout << "mitchell_07\n";
   99 : cout << "  Normal end of execution.\n";
  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.
times: compile 0.009903s, execution 0.011953s,  mpirank:0
 CodeAlloc : nb ptr  3619,  size :477800 mpirank: 0
Ok: Normal End