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 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 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 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 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
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.
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