fem2d_heat_rectangle, a FORTRAN90 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, over a rectangular region with a uniform grid.

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 code allows the user to supply two routines:

There are a few variables that are easy to manipulate. In particular, the user can change the variables NX and NY in the main code, 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 code.

The code 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 code 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.


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


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

Related Data and codes:

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

FEM2D_HEAT, a FORTRAN90 code which uses the finite element method and the backward Euler method to solve the 2D time-dependent heat equation on an arbitrary triangulated region.


FEM2D_POISSON_RECTANGLE, a FORTRAN90 code which solves Poisson's equation on a triangulated square, using the finite element method.


Janet Peterson.


  1. Hans Rudolf Schwarz,
    Finite Element Methods,
    Academic Press, 1988,
    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 08 July 2020.