Numerical methods for PDEs 1
2016/17 – problem set n. 6
Practical part for 18.11.2016
6.1. Recursive bisection with ALBERTA. Using the library ALBERTA, write a program that
• constructs a macro-triangulation consisting of the two triangles
conv{±e1 , e2 } and conv{±e1 , −e2 };
• recursively refines the triangle conv{±e1 , e2 } and one of its children;
• outputs to a file the resulting mesh (in the same format of as the
macro-triangulation).
Theoretical part for 21.10.2016
6.2. Surface distributions. Let Ω ⊂ Rd be a domain, ω ⊂ Ω a
subdomain with Lipschitz boundary and f ∈ L1 (∂ω), where L1 (∂ω)
indicates the set of all functions that are integrable with respect to
the surface measure of ∂ω (which coincides with the restriction of the
(d − 1)-dimensional Hausdorff-measure to ∂ω). Consider
�
�Tf , ϕ� =
fϕ
∂ω
C0∞ (Ω)
for ϕ ∈
and verify that it is a distribution. Moreover determine
its order and its support.
6.3. Distributional derivative of piecewise smooth functions.
Let
M : 0 = x0 < x1 < · · · < xn−1 < xn = 1
be a mesh of [0, 1] and v : [0, 1] → R a function such that
vi := v|]xi−1 ,xi [ ∈ C 1 [xi−1 , xi ].
Verify that the distributional derivative of v is
v� =
n
�
vi� χi +
i=1
n−1
�
[v](xi )δxi ,
i=1
where χi is the characteristic function of ]xi−1 , xi [ and we write
[v](xi ) := lim v(xi + δ) − v(xi − δ)
δ�0
for the jump of v in xi .
6.4. Friedrichs inequality. Let 1 ≤ p ≤ ∞, s ∈ N and Ω ⊂ Rd be
open and non-empty. Prove that, for all v ∈ W0s,p (Ω), we have
�v�0,p;Ω ≤ diam(Ω)s |v|s,p;Ω .
Hint: Generalize the partial proof in class.
6.5. Weak Laplacian. Let Ω be a domain of Rd . Prove that, for any
v ∈ H 1 (Ω), the distributional Laplacian given by
∀ϕ ∈ C0∞ (Ω)
�−Δv, ϕ� = �v, −Δϕ�
is (has a unique extension) in H −1 (Ω).
Information:
Course homepage: users.mat.unimi.it/users/veeser/mnedp1.html
Prof. A. Veeser
Office: 2051 (nel “sottotetto”)
Phone: 02.503.16186
Email: [email protected]
Office hours: Tuesday 9:30 – 11:30
Dott.ssa F. Fierro
Office: 2044
Phone: 02.503.16179
Email: francesca.fi[email protected]
Office hours: Friday 10:30 – 12:30