Suppose we want to solve the diffusion equation in the limit where @u/@t – 0. This gives rise to the

Suppose we want to solve the diffusion equation in the limit
where @u/@t – 0. This gives rise to the Poisson equation:

There is no initial condition associated with this equation,
because there is no evolution in time – the unknown function u is just a
function of space:  However, the equation is associated with
boundary conditions, say,

(a) Replace the second-derivative by a finite difference
approximation. Explain that this gives rise to a (tridiagonal) linear system,
exactly as for the implicit backward Euler scheme.

(b) Compare the equations from (a) with the equations
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Suppose we want to solve the diffusion equation in the limit
where @u/@t – 0. This gives rise to the Poisson equation:

There is no initial condition associated with this equation,
because there is no evolution in time – the unknown function u is just a
function of space:  However, the equation is associated with
boundary conditions, say,

(a) Replace the second-derivative by a finite difference
approximation. Explain that this gives rise to a (tridiagonal) linear system,
exactly as for the implicit backward Euler scheme.

(b) Compare the equations from (a) with the equations
generated by the backward Euler scheme. Show that the former arises in the
limit as in the latter.

(c) Construct an analytical solution of the Poisson equation
when f is constant.

 (d) The result from (b) tells us that we can take one very
long time step in a program implementing the backward Euler scheme and then
arrive at the solution of the Poisson equation. Demonstrate, by using a
program, that this is the case for the test problem from (c).

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