Finite Element Solution of the Heat Equation
on a Triangulated Region

FEM2D_HEAT is a C++ program which applies the finite element method to solve a form of the time-dependent heat equation over an arbitrary triangulated region.

The computational region is initially unknown by the program. The user specifies it by preparing a file containing the coordinates of the nodes, and a file containing the indices of nodes that make up triangles that form a triangulation of the region.

Normally, the user does not type in this information by hand, but has a program fill in the nodes, and perhaps another program that constructs the triangulation. However, in the simplest case, the user might construct a very crude triangulation by hand, and have TRIANGULATION_REFINE refine it to something more reasonable.

For the following ridiculously small example:

        |\   |\
        | \  | \
        6  7 8  9
        |   \|   \
the node file would be:
         0.0  0.0
         1.0  0.0
         2.0  0.0
         3.0  0.0
         4.0  0.0
         0.0  1.0
         1.0  1.0
         2.0  1.0
         3.0  1.0
         0.0  2.0
         1.0  2.0
         2.0  2.0
and the triangle file would be
         1  3 10  2  7  6
         3  5 12  4  9  8
        12 10  3 11  7  8

The program is set up to handle the time dependent heat equation with a right hand side function, and nonhomogeneous Dirichlet boundary conditions. The state variable U(T,X,Y) is then constrained by:

        Ut - ( Uxx + Uyy ) + K(x,y,t) * U = F(x,y,t)  in the region
                                        U = G(x,y,t)  on the boundary
                                        U = H(x,y,t)  at initial time TINIT.

To specify the right hand side function F(x,y,t), the linear coefficient K(x,y,t), the boundary condition function G(x,y,t), and the initial condition H(x,y,t), the user has to supply a file, perhaps called myprog.C, containing several functions:

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.


g++ fem2d_heat.cpp myprog.cpp
mv a.out fem2d_heat
fem2d_heat prefix
where prefix is the common file prefix:


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


FEM2D_HEAT is available in a C++ version and a FORTRAN90 version and a MATLAB version.

Related Programs:

FD2D_HEAT_STEADY, a C++ program which uses the finite difference method (FDM) to solve the steady (time independent) heat equation in 2D.

FEM1D_HEAT_STEADY, a C++ program which uses the finite element method to solve the steady (time independent) heat equation in 1D.

FEM2D_HEAT_SQUARE, a C++ library which defines the geometry of a square region, as well as boundary and initial conditions for a given heat problem, and is called by FEM2D_HEAT as part of a solution procedure.

STOCHASTIC_HEAT2D, a C++ program which implements a finite difference method (FDM) for the steady (time independent) 2D heat equation, with a stochastic heat diffusivity coefficient, using gnuplot to illustrate the results.


  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:

List of Routines:

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

Last revised on 13 November 2006.