The matrix equations for the Poisson problem do not involve time
and can be solved directly using either direct or iterative methods
for solving systems of equations. You have done that in your
introductory course on finite elements.
For the time-dependent heat equation, a few extra steps are needed.
This is because the equations we have developed so far still have
continuous time derivatives which need to be approximated.
Recall equations (45)
These equations are a coupled system of first-order ordinary differential equations rather than a system of algebraic equations.
One way of solving the system of differential equations
(45) is to use the generalized midpoint rule.
Substitute equation (48) into the first equation in
(46) to get
Collect terms containing and rearrange to get
We can solve equation (49) for . Substitute
this solution into equation (48) to get .
This approach is called the -form because the "velocity" or rate of change
of the unknown quantity is calculated first followed by the actual
quantity.
The usual engineer's approach is to stop after a solution has been
obtained and assume that this solution is adequate. However, it is
quite important to have some information about the quality of the
approximation. Unless such information is available, the finite
element solution is essentially useless because it could have little
resemblance to the actual solution.
Verification is the process of determining if the numerical
approximation
is an accurate representation of the mathematical model. The first
step in the process is to obtain a qualitative estimate of the
error in the approximation. Functional analysis plays a vital role
in determining these estimates of error.
The next step in the verification process is to obtain some information
about whether an approximate solution converges to the exact
solution as the mesh is refined. We can also determine what the
rate of convergence or order of accuracy for a particular
approach is. We will not get into the details of error estimation in this
course except for a few specific cases.
The final step in the verification process involves comparisons of
numerical results with known exact solutions and experimental results
of well-characterized benchmark problems.
We also need to validate our models. Validation is the process of
determining the degree to which our mathematical model represents
physical reality (as far as the intended use of the model is concerned).
Later, we will discuss some aspects of verification and validation in
the context of multi-physics problems.