# fem2d_poisson_rectangle_linear

fem2d_poisson_rectangle_linear, a C++ code which solves the 2D Poisson equation using the finite element method with piecewise linear triangular elements.

The computational region is a rectangle, with Dirichlet boundary conditions applied along the boundary. The state variable U(X,Y) is then constrained by:

```        - ( Uxx + Uyy ) = F(x,y)  in the region
U(x,y) = G(x,y)  on the region boundary
```

The computational region is first covered with an NX by NY rectangular array of points, creating (NX-1)*(NY-1) subrectangles. Each subrectangle is divided into two triangles, creating a total of 2*(NX-1)*(NY-1) geometric "elements".

We now assume that the unknown function U(x,y) can be represented as a linear combination of the basis functions associated with each node. For each node I, we determine a basis function PHI(I)(x,y), and evaluate the following finite element integral:

```        Integral ( Ux(x,y) * PHIx(I)(x,y) + Uy(x,y) * PHIy(I)(x,y) ) =
Integral ( F(x,y) * PHI(I)(x,y)
```
The set of all such equations yields a linear system for the coefficients of the representation of U.

### Languages:

fem2d_poisson_rectangle_linear is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

### Related Data and Programs:

FEM2D_POISSON_RECTANGLE, a C++ code which solves the 2D Poisson equation on a rectangle, using the finite element method, and piecewise quadratic triangular elements.

### 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

### Source Code:

Last revised on 05 March 2020.