Information sheet for Exam 1, MATH 610, Spring 2015
Notes: The exam will take place in class on Friday, March 27. No calculators, notes, textbooks
or other aids will be allowed. You will have the entire class period to work on the exam.
There will be no makeup exam unless it is for a pre-excused absence or a legitimate emergency.
The exam will cover lecture material through Monday, March 23. As a general rule, proofs,
definitions, examples, calculations, etc. that appeared in the lecture or on the theory homework
are all fair game. As a general rule, problems of the type and difficulty encountered in the
homework are a reasonable guide to the type of exam questions I might ask, obviously with the
brevity of the exam period taken into account.
Here is an outline of the main topics that have appeared in the course.
1. Function spaces (especially H 1 , H01 , L2 , and H k ); Poincaré inequalities. You should be
familiar with proofs of a Friedrichs–Poincaré
inequality of the type kukL2 (Ω) ≤ Ck∇ukL2 (Ω)
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when u ∈ H01 (Ω), but not when Ω udx = 0 (since as we mentioned in class the latter
proof uses more advanced tools). Also, when proving the Poincaré inequality in class for
u ∈ H01 (Ω), I said verbally but did not write down that completing the proof requires a
density argument. That is, the steps that we took in class apply for u smooth enough, say,
in C01 (Ω). C01 (Ω) is dense in H01 (Ω), so if u ∈ H01 (Ω) but not in C01 (Ω), we can approximate
u with a sequence {uk } which converges to u in H01 (Ω). Because both sides of the Poincaré
inequality are bounded by the H 1 norm and the Poincaré inequality holds for the sequence,
it must hold also for the limit.
2. Strong and weak (variational) form of PDEs; solvability and stability of variational problems via the Lax-Milgram theory.
3. Galerkin methods; error estimates via Cea’s Lemma.
4. Finite element methods for second-order elliptic problems in 1D and 2D. Error estimates
in H 1 , and in L2 by a duality argument when the given problem has the H 2 regularity
property.
5. The finite element triple. Unisolvence of finite element degrees of freedom.
6. Barycentric coordinates; definition of element bases via barycentric coordinates; nodal
bases.
7. The Bramble-Hilbert Lemma; proof of approximation properties for the Lagrange interpolant; reference elements and reference mapping arguments.
8. Assembly of finite element systems: Mass, stiffness, and system matrices; load vector.
Element system matrices and load vectors and their assembly into the global system.
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