# FEM1D_BVP_QUADRATIC Finite Element Method, 1D, Boundary Value Problem, Piecewise Quadratic Elements

FEM1D_BVP_QUADRATIC is a FORTRAN77 program which applies the finite element method, with piecewise quadratic 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 quadratoc 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 three point Gauss quadrature formula is used to estimate the resulting integrals over each interval.

### Languages:

FEM1D_BVP_QUADRATIC 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 FORTRAN77 program which applies the finite difference method to a two point boundary value problem in one spatial dimension.

FEM1D, a FORTRAN77 program which applies the finite element method to a linear two point boundary value problem in a 1D region.

FEM1D_ADAPTIVE, a FORTRAN77 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_LINEAR, is a FORTRAN77 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.

FEM1D_HEAT_STEADY, a FORTRAN77 program which uses the finite element method to solve the steady (time independent) heat equation in 1D.

FEM1D_NONLINEAR, a FORTRAN77 program which applies the finite element method to a nonlinear two point boundary value problem in a 1D region.

FEM1D_PMETHOD, a FORTRAN77 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_QUADRATIC, a FORTRAN77 program which applies the finite element method (FEM), with piecewise quadratic 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

### List of Routines:

• FEM1D_BVP_QUADRATIC solves a two point boundary value problem.
• H1S_ERROR_QUADRATIC: seminorm error of a finite element solution.
• L1_ERROR estimates the l1 error norm of a finite element solution.
• L2_ERROR_QUADRATIC estimates the L2 error norm of a finite element solution.
• R8MAT_SOLVE2 computes the solution of an N by N linear system.
• R8VEC_EVEN returns an R8VEC of evenly spaced values.
• TIMESTAMP prints out the current YMDHMS date as a timestamp.

You can go up one level to the FORTRAN77 source codes.

Last revised on 17 June 2014.