# FEM2D_PROJECT_FUNCTION Piecewise Linear Finite Element Approximation

FEM2D_PROJECT_FUNCTION is a C++ program which projects a function W(X,Y), given as a formula, into a given finite element space of piecewise linear triangular elements.

The result is that the function W(x,y) is approximated by a finite element function U(x,y) formed using piecewise linear triangular elements.

The computational region is a rectangle, which is divided up into a mesh of triangles using a grid of NX by NY points. For node K at grid point (I,J) in the interior, the associated basis function Vk(x,y) is used to pose the equation:

```        Integral U(x,y) Vk(x,y) dx = Integral W(x,y) Vk(x,y) dx
```
while, for node K at grid point (I,J) on the boundary, the associated degree of freedom is determined by the boundary condition
```        U(x,y) = W(x,y),
```

The conditions define a linear system for the coefficients Uk in the finite element expansion of U(x,y):

```        U(x,y) = sum ( 1 <= K <= M*N ) Uk * Vk(x,y)
```

The program computes these coefficients, compares U and W pointwise at the nodes, and computes the L2 norms of U, W and U-W.

### Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

### Languages:

FEM2D_PROJECT_FUNCTION is available in a C++ version and a FORTRAN90 version and a MATLAB version.

### Related Data and Programs:

FEM1D_PROJECT, a C++ program which projects data into a finite element space, including the least squares approximation of data, or the projection of a finite element solution from one mesh to another.

FEM2D, a data directory which contains examples of 2D FEM files, three text files that describe a 2D finite element geometry;

FEM2D_HEAT, a C++ program which solves the time dependent heat equation in the unit square.

FEM2D_PACK, a C++ library which includes utitlies for 2D finite element calculations.

FEM2D_POISSON, a C++ program which solves the 2D Poisson equation on a rectangle, using the finite element method, and piecewise quadratic triangular elements.

FEM2D_PROJECT, a C++ program which projects a function F(X,Y), given as a data, into a given finite element space of piecewise linear triangular elements.

FEM2D_SAMPLE, a C++ library which evaluates a finite element function defined on an order 3 or order 6 triangulation.

HOT_PIPE, a MATLAB program which uses FEM_50_HEAT to solve the heat equation for a pipe model.

HOT_POINT, a MATLAB program which uses FEM_50_HEAT to solve the heat equation for a 2D region with a hot point.

TRIANGULATION_ORDER3, a data directory which contains a description and examples of how an order 3 (linear) triangulation is defined by a node and element file.

TRIANGULATION_ORDER3_CONTOUR, a MATLAB program which makes contour plot of data defined on an order 3 triangulation.

### Reference:

1. Hans Rudolf Schwarz,
Finite Element Methods,
ISBN: 0126330107,
LC: TA347.F5.S3313.
2. Gilbert Strang, George Fix,
An Analysis of the Finite Element Method,
Cambridge, 1973,
ISBN: 096140888X,
LC: TA335.S77.
3. Olgierd Zienkiewicz,
The Finite Element Method,
Sixth Edition,
Butterworth-Heinemann, 2005,
ISBN: 0750663200,
LC: TA640.2.Z54

### List of Routines:

• MAIN is the main routine of the finite element program FEM2D_PROJECT_FUNCTION.
• EXACT calculates the exact solution.
• R8GE_FS factors and solves a R8GE system.

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

Last revised on 02 June 2009.