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

FEM1D_BVP_LINEAR, a Python 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:

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

• 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 values at the nodes, which can also be regarded as the finite element coefficients.

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:

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

FEM1D_BVP_QUADRATIC, a Python 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.

FEM2D_BVP_LINEAR, a Python 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:

• barebones.py, a version of fem1d_bvp_linear.py with the comments removed.
• fem1d_bvp_linear.py, sets up and solves the finite element problem.
• fem1d_bvp_linear_test00.py
• fem1d_bvp_linear_test01.py
• fem1d_bvp_linear_test02.py
• fem1d_bvp_linear_test03.py
• fem1d_bvp_linear_test04.py
• fem1d_bvp_linear_test05.py
• fem1d_bvp_linear_test06.py
• fem1d_bvp_linear_test07.py
• fem1d_bvp_linear_test08.py
• fem1d_bvp_linear_test09.py
• fem1d_bvp_linear_test10.py
• h1s_error_linear.py, estimates the seminorm of the error, given the piecewise linear solution of the finite element problem, and a function that evaluates the derivative of the exact solution.
• l1_error.py, estimates the little l1 norm of the error, given the solution of the finite element problem, and a function that evaluates the exact solution.
• l2_error_linear.py, estimates the L2 norm of the error, given the piecewise linear solution of the finite element problem, and a function that evaluates the exact solution.
• max_error_linear.py, estimates the max norm of the error, given the piecewise linear solution of the finite element problem, and a function that evaluates the exact solution.
• timestamp.py, prints the YMDHMS date as a timestamp.

Examples and Tests:

• fem1d_bvp_linear_test.py, calls all the tests.
• fem1d_bvp_linear_test.sh, runs all the tests.
• fem1d_bvp_linear_test.txt, the output file.

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