FEM1D_BVP_LINEAR
Finite Element Method, 1D, Boundary Value Problem, Piecewise Linear Elements

FEM1D_BVP_LINEAR is a FORTRAN90 program which applies the finite element method, with piecewise linear elements, to a two point boundary value problem in one spatial dimension, and compares the computed and exact solutions with the L2 and seminorm errors.

The boundary value problem (BVP) that is to be solved has the form:

        - d/dx ( a(x) * du/dx ) + c(x) * u(x) = f(x)
      

Boundary conditions are applied at the endpoints, and in this case, these are assumed to have the form:

        u(0.0) = 0.0;
        u(1.0) = 0.0.
      

To compute a finite element approximation, a set of n equally spaced nodes is defined from 0.0 to 1.0, a set of piecewise linear basis functions is set up, with one basis function associated with each node, and then an integral form of the BVP is used, in which the differential equation is multiplied by each basis function, and integration by parts is used to simplify the integrand.

A simple two point Gauss quadrature formula is used to estimate the resulting integrals over each interval.

Usage:

call fem1d_bvp_linear ( n, a, c, f, x, u )

• n is the number of equally spaced nodes.
• a is the function which evaluates a(x);
• c is the function which evaluates c(x);
• f is the function which evaluates f(x).
• x is the input vector of n nodes.
• u is the output vector of n finite difference values.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

FEM1D_BVP_LINEAR is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

FD1D_BVP, a FORTRAN90 program which applies the finite difference method to a two point boundary value problem in one spatial dimension.

FEM1D, a data directory which contains examples of 1D FEM files, three text files that describe a 1D finite element model;

FEM1D, a FORTRAN90 program which applies the finite element method to a linear two point boundary value problem in a 1D region.

FEM1D_ADAPTIVE, a FORTRAN90 program which applies the finite element method to a linear two point boundary value problem in a 1D region, using adaptive refinement to improve the solution.

FEM1D_BVP_QUADRATIC, a FORTRAN90 program which applies the finite element method (FEM), with piecewise quadratic elements, to a two point boundary value problem (BVP) in one spatial dimension, and compares the computed and exact solutions with the L2 and seminorm errors.

FEM1D_HEAT_STEADY, a FORTRAN90 program which uses the finite element method to solve the steady (time independent) heat equation in 1D.

FEM1D_NONLINEAR, a FORTRAN90 program which applies the finite element method to a nonlinear two point boundary value problem in a 1D region.

FEM1D_PMETHOD, a FORTRAN90 program which applies the p-method version of the finite element method to a linear two point boundary value problem in a 1D region.

FEM2D_BVP_LINEAR, a FORTRAN90 program which applies the finite element method (FEM), with piecewise linear elements, to a 2D boundary value problem (BVP) in a rectangle, and compares the computed and exact solutions with the L2 and seminorm errors.

Reference:

1. Dianne O'Leary,
   Finite Differences and Finite Elements: Getting to Know You,
   Computing in Science and Engineering,
   Volume 7, Number 3, May/June 2005.
2. Dianne O'Leary,
   Scientific Computing with Case Studies,
   SIAM, 2008,
   ISBN13: 978-0-898716-66-5,
   LC: QA401.O44.
3. Hans Rudolf Schwarz,
   Finite Element Methods,
   Academic Press, 1988,
   ISBN: 0126330107,
   LC: TA347.F5.S3313..
4. Gilbert Strang, George Fix,
   An Analysis of the Finite Element Method,
   Cambridge, 1973,
   ISBN: 096140888X,
   LC: TA335.S77.
5. Olgierd Zienkiewicz,
   The Finite Element Method,
   Sixth Edition,
   Butterworth-Heinemann, 2005,
   ISBN: 0750663200,
   LC: TA640.2.Z54

Source Code:

• fem1d_bvp_linear.f90, the source code.

Examples and Tests:

• fem1d_bvp_linear_test.f90, a sample calling program.
• fem1d_bvp_linear_test.txt, the output file.

One of the tests makes convergence plots in the H1, L2 and Max norms.

• h1.png
• l2.png
• mx.png

List of Routines:

• FEM1D_BVP_LINEAR solves a two point boundary value problem.
• GET_UNIT returns a free FORTRAN unit number.
• H1S_ERROR_LINEAR: seminorm error of a finite element solution.
• L1_ERROR estimates the l1 error norm of a finite element solution.
• L2_ERROR_LINEAR estimates the L2 error norm of a finite element solution.
• MAX_ERROR_LINEAR estimates the max error norm of a finite element solution.
• R8MAT_SOLVE2 computes the solution of an N by N linear system.
• R8VEC_LINSPACE creates a vector of linearly spaced values.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN90 source codes.