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

FEM1D_BVP_LINEAR is a MATLAB 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:

u = fem1d_bvp_linear ( n, @a, @c, @f, x )
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 values at the nodes, which can also be regarded as the finite element coefficients.

### 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:

BVP4C, MATLAB programs which illustrate how to use the MATLAB command bvp4c(), which can solve boundary value problems (BVP's) in one spatial dimension.

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

FEM_NEUMANN, a MATLAB program which sets up a time-dependent reaction-diffusion equation in 1D, with Neumann boundary conditions, discretized using the finite element method.

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

FEM1D_ADAPTIVE, a MATLAB 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 MATLAB 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_DISPLAY, a MATLAB program which reads three files defining a 1D arbitrary degree finite element function, and displays a plot.

FEM1D_FUNCTION_10_DISPLAY, a MATLAB program which reads a prefix defining three finite element data files, reads the data, samples the finite element function, and displays a plot.

FEM1D_LAGRANGE, a MATLAB library which sets up the matrices and vectors associated with the finite element method (FEM) solution of a boundary value problem (BVP) -u''+u=f(x), using Lagrange basis polynomials.

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

FEM1D_PMETHOD, a MATLAB 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 MATLAB 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

### Source Code:

• fem1d_bvp_linear.m, sets up and solves the finite element problem.
• h1s_error_linear.m, estimates the H1 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.m, 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.m, 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.m, 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.m, prints the YMDHMS date as a timestamp.

### Examples and Tests:

FEM1D_BVP_TEST runs example problems described by Dianne O'Leary, and several others.

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

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

Last revised on 11 July 2015.