Range closedness and the inf-sup conditions
04/09/2013
Notes by Ibrahim Al Balushi
1 Necessity and suciency for invertibility
In this section we derive a sucient and necessary condition for existence of an
inverse of a bounded linear map on Banach spaces.
MATH 765 ASSIGNMENT 1
DUE FRIDAY SEPTEMBER 20
1. Let X be a Hilbert space, and let A : X X be a bounded linear operator. Suppose
that A is strictly coercive in the sense that
Au, u u
2
for u X,
with some constant > 0.
a) Show that for any f X , there exis
Lecture 2
Notes on Finite Element Methods
Lecture 2
Pertrov-Galerkin methods
06/09/2013
Notes by Ibrahim Al Balushi
1 Applications to the Poisson problem
In this section we will apply some of the previous analysis regarding bilinear forms
to the Poisson p
MATH 765 ASSIGNMENT 2
DUE WEDNESDAY OCTOBER 9
1. Let T be an innite collection of triangles. For any triangle T , we let h = diam( ),
| | denote the area of , and let be the radius of the inscribed circle of . Show that
the followings are equivalent.
a) T
Lecture 5
Notes on Finite Element Methods
Lecture 5
Polynomial approximation in Sobolev spaces
1 Piecewise polynomial spaces
We dene the spaces of piecewise polynomials with respect to a partition P
S m,r (P ) = cfw_u C r () : u| S( ).
(5.1)
where S( ) =
Lecture 4
Notes on Finite Element Methods
Lecture 4
Characterization of nite element methods
There are two main characteristics of nite element methods (FEM):
Petrov-Galerkin approach.
Element-by-element computation.
Local nodal basis leading to sparse
Lecture 8
Notes on Finite Element Methods
Lecture 8
Lp-stability
Let be a bounded polyhedral domain in Rn , let P be a conforming partition
of and let SP = S d (P ) be a Lagrange nite element space. Write
hmin = min h ,
P
hmax = max h ,
= max .
P
P
Th
Lecture 10
Notes on Finite Element Methods
Lecture 10
More on real interpolation spaces
1 Technical facts
Let f X0 . Suppose that | |X1 is a semi-norm on X1 , i.e. X1 = X0 + | |X1 .
Recall the K -functional denition (9.1) which we write as K . We re-dene
Lecture 7
Notes on Finite Element Methods
Lecture 7
Direct and inverse estimates and Mesh
renement
1 Direct (Jackson) estimates
Let be star-shaped with respect to a ball, h = diam and let u W m,p ().
We have shown
n
u Qm uL () Chm p |u|W m,p () ,
(m > n,
Lecture 3
Notes on Finite Element Methods
Lecture 3
Strictly coercive problems
In the previous lectures we have established that in the Banach space setting, a
bounded linear map A : X Y is surely invertible provided it, as well as its
adjoint, are bounde
Lecture 9
Notes on Finite Element Methods
Lecture 9
Peetres K -method of interpolation
1 Introduction
Let X1 and X0 be Banach spaces with X1 X0 . We aim to construct intermediate spaces between X0 and X1 with favourable properties. We note that
it possibl
Lecture 6
Notes on Finite Element Methods
Lecture 6
Integration error for Lagrange nite elements
1 Bramble-Hilbert lemma
We will rst prove the Sobolev lemma.
Theorem 6.1 (Sobolev lemma). Let be a nite union of star-shaped domains.
Then W m,p () C() if m >