function [ a, f ] = assemble_poisson ( node_num, node_xy, ...
  element_num, element_node, quad_num, ib )

%% ASSEMBLE_POISSON assembles the system for the Poisson equation.
%
%  Discussion:
%
%    The matrix is known to be banded.  A special matrix storage format
%    is used to reduce the space required.  Details of this format are
%    discussed in the routine DGB_FA.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    22 July 2007
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, integer NODE_NUM, the number of nodes.
%
%    Input, real NODE_XY(2,NODE_NUM), the
%    coordinates of nodes.
%
%    Input, integer ELEMENT_NUM, the number of elements.
%
%    Input, integer ELEMENT_NODE(3,ELEMENT_NUM);
%    element_NODE(I,J) is the global index of local node I in element J.
%
%    Input, integer QUAD_NUM, the number of quadrature points used in assembly.
%
%    Input, integer IB, the half-bandwidth of the matrix.
%
%    Output, real A(3*IB+1,NODE_NUM), the NODE_NUM by NODE_NUM
%    coefficient matrix, stored in a compressed format.
%
%    Output, real F(NODE_NUM), the right hand side.
%
%  Local parameters:
%
%    Local, real BI, DBIDX, DBIDY, the value of some basis function
%    and its first derivatives at a quadrature point.
%
%    Local, real BJ, DBJDX, DBJDY, the value of another basis
%    function and its first derivatives at a quadrature point.
%

%
%  Initialize the arrays to zero.
%
  f(1:node_num) = 0.0;
  a(1:3*ib+1,1:node_num) = 0.0;
%
%  Get the quadrature weights and nodes.
%
  [ quad_w, quad_xy ] = quad_rule ( quad_num );
%
%  Add up all quantities associated with the element-th element.
%
  for element = 1 : element_num
%
%  Make a copy of the element.
%
    t3(1:2,1:3) = node_xy(1:2,element_node(1:3,element));
%
%  Map the quadrature points QUAD_XY to points PHYS_XY in the physical element.
%
    phys_xy(1:2,1:quad_num) = reference_to_physical_t3 ( t3, quad_num, quad_xy );

    area = abs ( triangle_area_2d ( t3 ) );

    w(1:quad_num) = quad_w(1:quad_num) * area;

    phys_rhs = rhs ( quad_num, phys_xy );

    phys_k = k_coef ( quad_num, phys_xy );
%
%  Consider the QUAD-th quadrature point..
%
    for quad = 1 : quad_num
%
%  Consider the TEST-th test function.
%
%  We generate an integral for every node associated with an unknown.
%  But if a node is associated with a boundary condition, we do nothing.
%
      for test = 1 : 3

        i = element_node(test,element);

        [ bi, dbidx, dbidy ] = basis_11_t3 ( t3, test, phys_xy(1:2,quad) );

        f(i) = f(i) + w(quad) * phys_rhs(quad) * bi;
%
%  Consider the BASIS-th basis function, which is used to form the
%  value of the solution function.
%
        for basis = 1 : 3

          j = element_node(basis,element);

          [ bj, dbjdx, dbjdy ] = basis_11_t3 ( t3, basis, phys_xy(1:2,quad) );

          a(i-j+2*ib+1,j) = a(i-j+2*ib+1,j) + w(quad) * ( ...
            dbidx * dbjdx + dbidy * dbjdy + phys_k(quad) * bj * bi );

        end

      end

    end

  end

  return
end