# fem2d_heat_rectangle

fem2d_heat_rectangle, a C++ code which solves the time-dependent 2D heat equation using the finite element method in space, and a method of lines in time with the backward Euler approximation for the time derivative.

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

```        Ut - ( Uxx + Uyy ) = F(x,y,t)  in the box;
U(x,y,t) = G(x,y,t) for (x,y) on the boundary;
U(x,y,t) = H(x,y,t) for t = t_init.
```

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". Because quadratic basis functions are to be used, each triangle will be associated not only with the three corner nodes that defined it, but with three extra midside nodes. If we include these additional nodes, there are now a total of (2*NX-1)*(2*NY-1) nodes in the region.

We now assume that, at any fixed time b, the unknown function U(x,y,t) can be represented as a linear combination of the basis functions associated with each node. The value of U at the boundary nodes is obvious, so we concentrate on the NUNK interior nodes where U(x,y,t) is unknown. For each node I, we determine a basis function PHI(I)(x,y), and evaluate the following finite element integral:

```        Integral ( Ux(x,y,t) * dPHIdx(I)(x,y) + dUdy(x,y,t) * dPHIdy(I)(x,y) ) =
Integral ( F(x,y,t) * PHI(I)(x,y)
```

The time derivative is handled by the backward Euler approximation.

The program allows the user to supply two routines:

• rhs ( x, y, time ) returns the right hand side F(x,y,time) of the heat equation.
• exact_u ( node_num, node_xy, time, u_exact ) returns the exact solution U_EXACT evaluated at each of the NODE_NUM points whose coordinates are stored in NODE_XY(1:2,1:NODE_NUM), at time TIME.

There are a few variables that are easy to manipulate. In particular, the user can change the variables NX and NY in the main program, to change the number of nodes and elements. The variables (XL,YB) and (XR,YT) define the location of the lower left and upper right corners of the rectangular region, and these can also be changed in a single place in the main program.

The program writes out a file containing an Encapsulated PostScript image of the nodes and elements, with numbers. Unfortunately, for values of NX and NY over 10, the plot is too cluttered to read. For lower values, however, it is a valuable map of what is going on in the geometry.

The program is also able to write out a file containing the solution value at every node. This file may be used to create contour plots of the solution.

### Licensing:

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

### Languages:

fem2d_heat_rectangle is available in a C++ version and a FORTRAN90 version and a MATLAB version.

### Related Data and Programs:

FEM2D_HEAT, a C++ code which solves the time dependent heat equation on an arbitrary triangulated region in 2D.

### 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.