fem1d_nonlinear/fem1d_nonlinear.html

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      FEM1D_NONLINEAR - Finite Element Method for 1D Nonlinear Problem
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      FEM1D_NONLINEAR <br> Finite Element Method for 1D Nonlinear Problem.
    </h1>

    <hr>

    <p>
      <b>FEM1D_NONLINEAR</b>
      is a FORTRAN90 program which
      applies the finite element method to a simple nonlinear
      boundary value problem in one spatial dimension.
    </p>

    <p>
      The nonlinear boundary problem involves an unknown function <b>u</b> on
      an interval <b>[a,b]</b>.  Typically, a single boundary condition, either
      <b>u(a)</b> or <b>u'(a)</b>, is given at the left endpoint, and
      <b>u(b)</b> or <b>u'(b)</b> at the right endpoint.  The associated differential
      equation, which must hold in the interior of <b>[a,b]</b>, happens to have
      the form:
      <pre>
        -d/dx ( p(x) du/dx ) + q(x) * u + u * du/dx = f(x)
      </pre>
      where <b>p(x)</b>, <b>q(x)</b> and <b>f(x)</b> are given functions.
    </p>

    <p>
      The nonlinearity arises because of the term <b>u*du/dx</b>; if this term
      were not present, standard finite element techniques would allow the system
      to be set up and solved almost as easily as a linear system of algebraic
      equations is solved.
    </p>

    <p>
      Newton's method, which works well for scalar nonlinear equations, can also
      be applied to this problem, as long as we are willing to extend our notions
      of a function and derivative.  We need to imagine our differential equation
      as a function <b>F(u)</b>:
      <pre>
        F(u) = -d/dx ( p(x) du/dx ) + q(x) * u + u * du/dx - f(x)
      </pre>
      (with the boundary conditions wrapped in here somewhere as well!)
      If we differentiate this function, we get a Jacobian operator, which is
      evaluated at <b>u</b>, and applied to any small increment <b>v</b>.  This equation
      implicitly describes the tangent plane of solutions near to a given
      solution <b>u</b>.
      <pre>
        J(u,v) = -d/dx ( p(x) dv/dx ) + q(x) * v + u * dv/dx + v * du/dx
      </pre>
      Now if we apply the finite element formulation to represent <b>u</b> and
      <b>v</b> in terms of sums of basis functions, we can set up a linear
      system, to be evaluated at <b>u</b> and solved for the Newton increment
      <b>delta_u</b>:
      <pre>
        J(u,delta_u) = - F(u)
      </pre>
      By using the Newton increment to update <b>u</b> and repeating the process
      as needed, we can expect to get a good finite element solution of our original
      nonlinear boundary value problem.
    <p>

    <h3 align = "center">
      Licensing:
    </h3>

    <p>
      The computer code and data files described and made available on this web page
      are distributed under
      <a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
    </p>

   <h3 align = "center">
      Languages:
    </h3>

    <p>
      <b>FEM1D_NONLINEAR</b> is available in
      <a href = "../../cpp_src/fem1d_nonlinear/fem1d_nonlinear.html">a C++ version</a> and
      <a href = "../../f77_src/fem1d_nonlinear/fem1d_nonlinear.html">a FORTRAN77 version</a> and
      <a href = "../../f_src/fem1d_nonlinear/fem1d_nonlinear.html">a FORTRAN90 version</a> and
      <a href = "../../m_src/fem1d_nonlinear/fem1d_nonlinear.html">a MATLAB version.</a>
    </p>

    <h3 align = "center">
      Related Data and Programs:
    </h3>

    <p>
      <a href = "../../data/fem1d/fem1d.html">
      FEM1D</a>,
      a data directory which
      contains examples of 1D FEM files,
      three text files that describe a 1D finite element model;
    </p>

    <p>
      <a href = "../../f_src/fem1d/fem1d.html">
      FEM1D</a>,
      a FORTRAN90 program which
      applies the finite element method to a 1D linear two point boundary value problem.
    </p>

    <p>
      <a href = "../../f_src/fem1d_adaptive/fem1d_adaptive.html">
      FEM1D_ADAPTIVE</a>,
      a FORTRAN90 program which
      applies the finite element method to a 1D linear two point boundary value problem
      using adaptive refinement to improve the solution.
    </p>

    <p>
      <a href = "../../f_src/fem1d_bvp_linear/fem1d_bvp_linear.html">
      FEM1D_BVP_LINEAR</a>,
      a FORTRAN90 program which
      applies the finite element method, with piecewise linear elements,
      to a two point boundary value problem in one spatial dimension.
    </p>

    <p>
      <a href = "../../f_src/fem1d_heat_steady/fem1d_heat_steady.html">
      FEM1D_HEAT_STEADY</a>,
      a FORTRAN90 program which
      uses the finite element method to solve the steady (time independent)
      heat equation in 1D.
    </p>

    <p>
      <a href = "../../f_src/fem1d_pack/fem1d_pack.html">
      FEM1D_PACK</a>,
      a FORTRAN90 library which
      contains utilities for 1D finite element calculations.
    </p>

    <p>
      <a href = "../../f_src/fem1d_pmethod/fem1d_pmethod.html">
      FEM1D_PMETHOD</a>,
      a FORTRAN90 program which
      applies the p-method version of the finite element method to a 1D linear
      two point boundary value problem.
    </p>

    <p>
      <a href = "../../f_src/fem1d_project/fem1d_project.html">
      FEM1D_PROJECT</a>,
      a FORTRAN90 program which
      projects data into a finite element space, including the least squares
      approximation of data, or the projection of a finite element solution
      from one mesh to another.
    </p>

    <p>
      <a href = "../../f_src/fem1d_sample/fem1d_sample.html">
      FEM1D_SAMPLE</a>,
      a FORTRAN90 program which
      samples a scalar or vector finite element function of one variable,
      defined by <a href = "../data/fem/fem.html">FEM files</a>,
      returning interpolated values at the sample points.
    </p>

    <h3 align = "center">
      Reference:
    </h3>

    <p>
      <ol>
        <li>
          Hans Rudolf Schwarz,<br>
          Finite Element Methods,<br>
          Academic Press, 1988,<br>
          ISBN: 0126330107,<br>
          LC: TA347.F5.S3313..
        </li>
        <li>
          Gilbert Strang, George Fix,<br>
          An Analysis of the Finite Element Method,<br>
          Cambridge, 1973,<br>
          ISBN: 096140888X,<br>
          LC: TA335.S77.
        </li>
        <li>
          Olgierd Zienkiewicz,<br>
          The Finite Element Method,<br>
          Sixth Edition,<br>
          Butterworth-Heinemann, 2005,<br>
          ISBN: 0750663200,<br>
          LC: TA640.2.Z54
        </li>
      </ol>
    </p>

    <h3 align = "center">
      Source Code:
    </h3>

    <p>
      <ul>
        <li>
          <a href = "fem1d_nonlinear.f90">fem1d_nonlinear.f90</a>, the source code.
        </li>
        <li>
          <a href = "fem1d_nonlinear.sh">fem1d_nonlinear.sh</a>,
          commands to compile the source code.
        </li>
      </ul>
    </p>

    <h3 align = "center">
      Examples and Tests:
    </h3>

    <p>
      <ul>
        <li>
          <a href = "problem1_output.txt">problem1_output.txt</a>,
          the output from solving problem 1.
        </li>
        <li>
          <a href = "problem2_output.txt">problem2_output.txt</a>,
          the output from solving problem 2.
        </li>
      </ul>
    </p>

    <h3 align = "center">
      List of Routines:
    </h3>

    <p>
      <ul>
        <li>
          <b>MAIN</b> is the main program for FEM1D_NONLINEAR.
        </li>
        <li>
          <b>ASSEMBLE_NEWTON</b> assembles the Newton linear system.
        </li>
        <li>
          <b>ASSEMBLE_PICARD</b> assembles the Picard linear system.
        </li>
        <li>
          <b>COMPARE</b> compares the computed and exact solutions.
        </li>
        <li>
          <b>FF</b> returns the right hand side of the differential equation.
        </li>
        <li>
          <b>GEOMETRY</b> sets up the geometry for the interval [XL,XR].
        </li>
        <li>
          <b>INIT</b> initializes variables that define the problem.
        </li>
        <li>
          <b>OUTPUT</b> prints out the computed solution at the nodes.
        </li>
        <li>
          <b>PHI</b> evaluates a linear basis function and its derivative.
        </li>
        <li>
          <b>PP</b> returns the value of the coefficient function P(X).
        </li>
        <li>
          <b>PRSYS</b> prints out the tridiagonal linear system to be solved.
        </li>
        <li>
          <b>QQ</b> returns the value of the coefficient function Q(X).
        </li>
        <li>
          <b>SOLVE</b> solves a tridiagonal matrix system of the form A*x = b.
        </li>
        <li>
          <b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
        </li>
        <li>
          <b>U_EXACT</b> returns the value of the exact solution at a point X.
        </li>
      </ul>
    </p>

    <p>
      You can go up one level to <a href = "../f_src.html">
      the FORTRAN90 source codes</a>.
    </p>

    <hr>

    <i>
      Last revised on 29 April 2007.
    </i>

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