function [ el2, eh1 ] = errors ( element_area, element_node, node_xy, u, ...
  element_num, nnodes, quad2_num, node_num, time )

%% ERRORS calculates the L2 and H1-seminorm errors.
%
%  Discussion:
%
%    This routine uses a 13 point quadrature rule in each element,
%    in order to estimate the values of
%
%      EL2(t) = Sqrt ( Integral ( U(x,y,t) - Uh(x,y,t) )**2 dx dy )
%
%      EH1(t) = Sqrt ( Integral ( Ux(x,y,t) - Uhx(x,y,t) )**2 + 
%                               ( Uy(x,y,t) - Uhy(x,y,t) )**2 dx dy )
%
%    Here U is the exact solution, and Ux and Uy its spatial derivatives,
%    as evaluated by a user-supplied routine.
%
%    Uh, Uhx and Uhy are the computed solution and its spatial derivatives,
%    as specified by the computed finite element solution.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    10 April 2006
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, real ELEMENT_AREA(ELEMENT_NUM), the area of elements.
%
%    Input, integer ELEMENT_NODE(NNODES,ELEMENT_NUM);
%    ELEMENT_NODE(I,J) is the global index of local node I in element J.
%
%    Input, real NODE_XY(2,NODE_NUM), the nodes.
%
%    Input, real U(NODE_NUM), the coefficients of the solution.
%
%    Input, integer ELEMENT_NUM, the number of elements.
%
%    Input, integer NNODES, the number of nodes used to form one element.
%
%    Input, integer QUAD2_NUM, the number of points in the quadrature rule.
%    This is actually fixed at 13.
%
%    Input, integer NODE_NUM, the number of nodes.
%
%    Input, real TIME, the current time.
%
%    Output, real EL2, the L2 error.
%
%    Output, real EH1, the H1 seminorm error.
%
%  Local Parameters:
%
%    Local, real AR, the weight for a given quadrature point
%    in a given element.
%
%    Local, real BB, BX, BY, a basis function and its first
%    derivatives evaluated at a particular quadrature point.
%
%    Local, real EH1, the H1 seminorm error.
%
%    Local, real EL2, the L2 error.
%
%    Local, real UEX, UEXX, UEXY, the exact solution and its first
%    derivatives evaluated at a particular quadrature point.
%
%    Local, real UH, UHX, UHY, the computed solution and its first
%    derivatives evaluated at a particular quadrature point.
%
%    Local, real WQE(NQE), stores the quadrature weights.
%
%    Local, real X, Y, the coordinates of a particular
%    quadrature point.
%
%    Local, real XQE(NQE), YQE(NQE), stores the location
%    of quadrature points in a given element.
%
  el2 = 0.0;
  eh1 = 0.0;

  for element = 1 : element_num

    [ wqe, xqe, yqe ] = quad_e ( node_xy, element_node, element, ...
      element_num, nnodes, node_num, quad2_num );

    for quad = 1 : quad2_num

      ar = element_area(element) * wqe(quad);
      x = xqe(quad);
      y = yqe(quad);

      uh = 0.0;
      dudxh = 0.0;
      dudyh = 0.0;

      for in1 = 1 : nnodes

        i = element_node(in1,element);

        [ bi, dbidx, dbidy ] = qbf ( x, y, element, in1, node_xy, ...
          element_node, element_num, nnodes, node_num );

        uh    = uh    + bi    * u(i);
        dudxh = dudxh + dbidx * u(i);
        dudyh = dudyh + dbidy * u(i);

      end

      xy(1,1) = x;
      xy(2,1) = y;

      [ u_exact, dudx_exact, dudy_exact ] = exact_u ( 1, xy, time );

      el2 = el2 + ar * ( uh - u_exact(1) )^2 ;

      eh1 = eh1 + ar * ( ( dudxh - dudx_exact(1) )^2 ...
                       + ( dudyh - dudy_exact(1) )^2 );

    end

  end

  el2 = sqrt ( el2 );
  eh1 = sqrt ( eh1 );

  fprintf ( 1, '  %12f  %12f  %12f\n', time, el2, eh1 );

  return
end