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

FEM1D_BVP_LINEAR is a C++ 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)
```
in the interval 0 < x < 1. The functions a(x), c(x), and f(x) are given.

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:

fem1d_bvp_linear ( n, a, c, f, x, u )
where
• 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.

### Languages:

FEM1D_BVP_LINEAR is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version and a Python version.

### Related Data and Programs:

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

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

FEM1D_ADAPTIVE, a C++ 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 C+= 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 C++ program which uses the finite element method to solve the steady (time independent) heat equation in 1D.

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

FEM1D_PMETHOD, a C++ 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 C++ 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,
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

### Examples and Tests:

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

### List of Routines:

• COMPUTE_L2_ERROR estimates the L2 error norm of a finite element solution.
• COMPUTE_SEMINORM_ERROR estimates the seminorm error of a finite element solution.
• FEM1D_BVP_LINEAR solves a two point boundary value problem.
• R8MAT_SOLVE2 computes the solution of an N by N linear system.
• R8VEC_EVEN returns an R8VEC of evenly spaced values.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the C++ source codes.

Last revised on 20 August 2010.