FEM2D_HEAT_RECTANGLE
Finite Element Solution in 2D
Time Dependent Heat Equation

FEM2D_HEAT_RECTANGLE is a MATLAB program 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:

• FUNCTION VALUE = RHS ( X, Y, T ) returns the right hand side F(x,y,t) of the heat equation.
• FUNCTION U_EXACT = EXACT_U ( X, Y, T ) returns the exact solution of the equation (assuming this is known.) This routine is necessary so that error analysis can be performed, reporting the L2 and H1 seminorm errors between the true and computed solutions.

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 GNU LGPL 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_RECTANGLE_STEADY_SPMD, a MATLAB program which uses the MATLAB Parallel Computing Toolbox in SPMD mode to set up and solve a distributed linear system for the steady 2D heat equation.

Reference:

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:

• fem2d_heat_rectangle.m, the source code.

Examples and Tests:

• fem2d_heat_rectangle.txt, the output file.
• fem2d_heat_rectangle_nodes.png, a PNG image of the nodes, for NX = NY = 7 (the picture is hard to read for larger values of NX and NY);
• nodes.txt, a text file containing a list, for each node, of its X and Y coordinates;
• elements.png, a PNG image of the elements, for NX = NY = 7 (the picture is hard to read for larger values of NX and NY);
• elements.txt, a text file containing a list, for each element, of the six nodes that compose it;
• time.txt, a text file containing the solution times;
• u0000.txt, the solution U at time step 0;
• u0001.txt, the solution U at time step 1;
• u0002.txt, the solution U at time step 2;
• u0003.txt, the solution U at time step 3;
• u0004.txt, the solution U at time step 4;
• u0005.txt, the solution U at time step 5;
• u0006.txt, the solution U at time step 6;
• u0007.txt, the solution U at time step 7;
• u0008.txt, the solution U at time step 8;
• u0009.txt, the solution U at time step 9;
• u0010.txt, the solution U at time step 10;

The MATLAB program CONTOUR_SEQUENCE4 can make contour plots from the sequence of solutions:

• u0000.png, the solution U at time step 0;
• u0001.png, the solution U at time step 1;
• u0002.png, the solution U at time step 2;
• u0003.png, the solution U at time step 3;
• u0004.png, the solution U at time step 4;
• u0005.png, the solution U at time step 5;
• u0006.png, the solution U at time step 6;
• u0007.png, the solution U at time step 7;
• u0008.png, the solution U at time step 8;
• u0009.png, the solution U at time step 9;
• u0010.png, the solution U at time step 10;

You can go up one level to the MATLAB source codes.