**Questions**

Below you will find some sample question that may help you better digest
the material for this course. You may find them useful preparing yourselves
for the quizzes. They should be particularly useful to those
who will choose *Graduate Numerical Analysis* as a part of their
Comprehensive Exam.

Discuss weak formulation of boundary value problems and their relation
to the corresponding classical formulations
Consider functions defined on a periodic domain.
How to calculate efficiently Sobolev norms of these functions when
the differentiability index is non-integer?
What alternatives exist as regards numerical solution of boundary
value problems with symmetric operators?
What is the difference between *conforming* and *non-conforming*
Finite Elements?
How can one increase the accuracy of the numerical solution obtained
with a Finite Element Method (h-refinement vs. p-refinement)?
Consider the problem addressed in the Example "fem_01.m" posted on the
course website. How would the properties of the system matrix change
if sin(k*Pi*x), k=1, ..., N were used as the basis functions?
What determines the *sparsity structure* of the "stiffness"
matrix in the Finite Element Method
What does it mean that a Finite Element Method is *conforming*?
You are given a simple, second-order boundary value problem in 1D
[e.g., -u''-u=f in (0,1)] with either Dirichlet or Neumann boundary conditions.
Propose a simple Finite Element Method that you could use in order to solve
this problem. Discuss how the boundary conditions in the problem
will affect the solution methodology.
Given a linear (in 1D), triangular / quadrilateral (in 2D) Finite Element,
show how to contract a system of C^0/C^1 Lagrangian basis functions
on this element
What are *Gauss quadratures*? Discuss their properties that you
think are important from the point of view of the Finite Element Method.
Discuss how the reference element technique can be used to evaluate
numerically the integrals which appear in the Finite Element Method.
What are the properties that must be satisfied by the PDE, so that
the Ritz-Galerkin method could be applied?
Why is *coercivity* important in the Finite Element solution
of Partial Differential Equations?
What additional precautions need to be taken when applying a
Finite Element Method to solve a first-order Partial / Ordinary
Differential Equations?