Chapter 6 A posteriori error estimates for nite
element approximations
6.1 Introduction
The a posteriori error estimation of nite element approximations of
elliptic boundary value problems has reached some state of maturity,
as it is documented by a varie
Chapter 1 Foundations of Elliptic Boundary Value
Problems
1.1 Euler equations of variational problems
Elliptic boundary value problems often occur as the Euler equations of
variational problems the latter representing the optimality conditions
of minimiza
Chapter 8 Curl-conforming edge element methods
8.1 Maxwells equations
8.1.1 Introduction
Electromagnetic phenomena can be described by the electric eld
E, the electric induction D, the current density J as well as
the magnetic eld H and the magnetic induc
Chapter 3 Conforming Finite Element Methods
3.1 Foundations
3.1.1 Ritz-Galerkin Method
Let V be a Hilbert space, a(, ) : V V lR a bounded, V-elliptic
bilinear form and : V lR a bounded linear functional. We want to
approximate the variational equation:
Fi
Chapter 4 A priori error estimates for conforming
nite element approximations
4.1 Interpolation in Sobolev spaces
For a simply-connected Lipschitz domain lRd and m lN0 we
consider the quotient space W m+1,p ()/Pm (), p [1, ], whose
elements are equivalenc
Chapter 7 Mixed nite element methods
7.1 The membrane problem revisited
We recall from Chapter 1 that the computation of the equilibrium
state of a clamped membrane amounts to the solution of the convex
minimization problem
(7.1)
J (u) =
inf
1
v H0 ()
J (
Chapter 8
Adaptive Finite Element Techniques
8.1 Introduction
The usual nite element analysis would proceed from the selection of a mesh and basis
to the generation of a solution to an accuracy appraisal and analysis. Experience is the
traditional method
Chapter 10
Hyperbolic Problems
10.1 Conservation Laws
We have successfully applied nite element methods to elliptic and parabolic problems
however, hyperbolic problems will prove to be more di cult. We got an inkling of this
while studying convection-di u
Chapter 9
Parabolic Problems
9.1 Introduction
The nite element method may be used to solve time-dependent problems as well as
steady ones. This e ort involves both parabolic and hyperbolic partial di erential systems. Problems of parabolic type involve di
Chapter 2 Sobolev spaces
In this chapter, we give a brief overview on basic results of the theory
of Sobolev spaces and their associated trace and dual spaces.
2.1 Preliminaries
Let be a bounded domain in Euclidean space lRd . We denote by
its closure an
Chapter 7
Analysis of the Finite Element
Method
7.1 Introduction
Finite element theory is embedded in a very elegant framework that enables accurate a
priori and a posteriori estimates of discretization errors and convergence rates. Unfortunately, a large
Chapter 5 A priori error estimates for nonconforming nite element approximations
5.1 Strangs rst lemma
We consider the variational equation
(5.1)
a(u, v ) =
v V H 1 () ,
(v ) ,
and assume that the conditions of the Lax-Milgram lemma are satised
so that (5