FEM2D_HEAT_SPARSE_SQUARE
A Problem in a Square Region for FEM2D_HEAT_SPARSE

FEM2D_HEAT_SPARSE_SQUARE is a MATLAB library which defines the geometry and other data of the square problem, so that a solution can be computed by fem2d_heat_sparse.

This mesh uses 81 nodes and forms 32 quadratic triangular elements.

We assume that the equation to be solved is

dUdT - Laplacian U + K * U = F

K = 0,

F = (2*pi*pi-1)*sin(pi*x)*sin(pi*y)*exp(-t).

U = sin(pi*x) * sin(pi*y) * exp(-t).

Licensing:

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

Related Data and Programs:

CONTOUR_SEQUENCE4, a MATLAB program which makes a sequence of contour plots based on a single XY coordinate file, and a sequence of Z(X,Y) data files.

FEM2D_HEAT_SPARSE, a MATLAB program which solves the time dependent heat equation in an arbitrary triangulated 2D region, using MATLAB's sparse matrix storage format and solver.

Source Code:

• dirichlet_condition.m, the user-supplied routine to evaluate the boundary conditions;
• initial_condition.m, the user-supplied routines to evaluate the initial conditions;
• k_coef.m, the user-supplied routines to evaluate the linear coefficient.
• rhs.m, the user-supplied routines to evaluate the right hand side;
• square_output.txt, output from a run of the program;
• square_nodes.png, a PNG image of the 49 nodes;
• square_nodes.txt, a text file containing a list, for each node, of its X and Y coordinates;
• square_elements.png, a PNG image of the 32 elements;
• square_elements.txt, a text file containing a list, for each element, of the six nodes that compose it;

Examples and Tests:

The following solution data was computed over 10 time steps:

• square_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 code page.