fem1d_heat_steady, a FORTRAN90 code which applies the finite element method (FEM), with piecewise linear elements, to solve the steady state heat equation in one spatial dimension.

The finite element method is used with piecewise linear elements.

The steady state heat equation that is to be solved has the form:

```        - d/dx ( k(x) * du/dx ) = f(x)
```
in the interval A < x < B. The functions k(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(A) = UA;
u(B) = UB.
```

To compute a finite element approximation, a set of N equally spaced nodes is defined from A to B, a set of piecewise linear basis functions is set up, with one basis function associated with each node, and then 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_heat_steady ( n, a, b, ua, ub, k, f, x, u )
where
• n is the number of equally spaced nodes.
• a, b are the left and right endpoints.
• ua, ub are the prescribed values of U at the endpoints;
• k is the function which evaluates k(x);
• f is the function which evaluates f(x).
• x is the input vector of n nodes, with x(1) = a, x(n) = b.
• u is the output vector containing the estimated solution value at the nodes.

### Languages:

fem1d_heat_steady is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

### Related Data and codes:

FD1D_HEAT_STEADY, a FORTRAN90 code which applies the finite difference method to the steady state heat equation 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 code which applies the finite element method to a linear two point boundary value problem in a 1D region.

FEM1D_ADAPTIVE, a FORTRAN90 code 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, a FORTRAN90 code which applies the finite element method to a linear two point boundary value problem in a 1D region, using piecewise linear elements.

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

FEM1D_PACK, a FORTRAN90 code which contains utilities for 1D finite element calculations.

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

STOCHASTIC_HEAT2D, a FORTRAN90 code which implements a finite difference method (FDM) for the steady (time independent) 2D heat equation, with a stochastic heat diffusivity coefficient.

### 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:

Last revised on 06 July 2020.