Information sheet for Final Exam, MATH 610, Spring 2015
Notes: The exam will take place in our regular classroom on Friday, May 8, at 10:30 a.m. No
calculators, notes, textbooks or other aids will be allowed. You will have 2 hours to complete
the exam. There will be no makeup exam unless it is for a pre-excused absence or a legitimate
emergency. The exam is comprehensive, potentially covering material from the entire semester
with the exception of the last three lectures concerning finite difference methods for hyperbolic
problems. 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.
Since the first exam, we have covered:
1. Quadrature errors and their analysis. Statement of Strang’s First Lemma and its proof.
Bounds for quadrature error terms appearing in Strang’s First Lemma by mapping to the
reference element and application of the Bramble-Hilbert Lemma.
2. Basic theory for parabolic initial-boundary value problems. Weak form. Energy estimates
that can be obtained by testing variously by u or ut .
3. Definition of spatially semi discrete finite element methods for parabolic problems in variational and matrix forms. Proof of stability by energy arguments. Proof of error estimates
in L2 and H 1 norms using Wheeler’s trick (the Ritz projection).
4. Fully discrete finite element methods for parabolic problems using the backward Euler
time discretization. Proof of error estimates in L2 by combining Wheeler’s trick with
unconditional stability of the method.
5. Definition of basic finite difference methods for the 1D heat equation on free space and
the unit interval (backward Euler, forward Euler, Crank-Nicholson). Proof of (conditional
or unconditional) stability in `∞,h for both Euler methods. Proof of error bounds in `∞,h
by combining analysis of the truncation error with stability.
Topics covered prior to the first exam are fair game for the final. As a review, here
is an outline of the main topics that appeared prior to the first exam:
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 ∈ H0 (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.
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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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