5520 MATH Finite Element Methods.
April 14, 2009
Assignment 4 (C)
Consider the nonhomogeneous Heat equation u(x, t) t x (x) u(x, t) x = f (x, t), x (0, 1), t > 0,
u(0, t) = u(1, t) = 0,
t > 0, x (0, 1),
u(x, 0) = g(x), where (x), f (x, t), and g(x) are some given functions.
1. Write down the weak form of the equation. What are the appropriate spaces of function? Explain. 2. Let Th be the partition 0 = x0 < x1 < x2 < < xN = 1 and Vh be the space of continuous piecewise linear functions on the partition Th . Dene the semidiscrete Galerkin solution uh (t) : [0, T ] Vh . 3. Dene fully discrete Backward Euler method. 4. Modify your code from HW 2 to solve the heat equation by backward Euler method. 5. Once you it, accomplish run the convergence tests with the exact solution u(x, t) = t2 sin (x), ( 1), by successive rening the mesh in space and time. Remember that now the error e(x, t) between the exact and backward Euler solution is a function of space and time. The theory says that if the solution is smooth the error at the nal time T should obey the estimate e(T )
L2 (0,1)
C h2 + k ,
where h is the mesh size in space and k is the time step. Si...