2 DGM for elliptic problems

2.3 DGM based on a primal formulation 19
iii) B s,σ
h
(u h,v h ) = l s,σ
h (v h) ∀v h ∈ S hp (2.62) A2.22c
iv) B n,σ
h
(u h,v h ) = l n,σ
h (v h) ∀v h ∈ S hp (2.63) A2.22d
v) B i,σ
h (u h,v h ) = l i,σ
h (v h) ∀v h ∈ S hp (2.64) A2.22e
where the forms Bh s, Bn h ,... and ls h ,ln h
,... are defined by (A2.20a 2.50) – ( 2.54) A2.20e and
( 2.55) A2.21a – ( 2.59), A2.21e respectively.
If we denote by B h any form defined by ( 2.50) A2.20a – ( 2.54) A2.20e and by l h we denote
the corresponding form given by ( 2.55) A2.21a – ( 2.59), A2.21e the discrete problem can be
formulated to find u h ∈ S hp satisfying the identity
B h (u h ,v h ) = l h (v h ) ∀v h ∈ S hp . (2.65) dispr
From the construction of the forms B and l one can see that the strong
solution u ∈ H 2 (Ω) of problem ( 2.1) A0.1 – ( 2.3) A0.2b satisfies the identity
terminology
B h (u,v) = l h (v) ∀v ∈ H 2 (Ω, T h ) (2.66) A2.23
which represents the consistency of the method. The expressions ( 2.65) dispr and
( 2.66) A2.23 imply the so-called Galerkin orthogonality of the error e h = u h − u of
the method:
B h (e h ,v h ) = 0 ∀v h ∈ S hp , (2.67) A2.23a
which will be used in the analysis of error estimates.
In contrast to standard conforming finite element techniques, both Dirichlet
and Neumann boundary conditions are included automatically in formulation
( 2.65) dispr of the discrete problem. This is an advantage particularly in
the case of nonhomogeneous Dirichlet boundary conditions, because it is not
necessary to construct subsets of finite element spaces formed by functions
approximating the Dirichlet boundary condition in a suitable way.
Remark 2.9. The method ( 2.60) A2.22a was introduced by Delves et al. ([DH79],
Delves1
Delves2
[DP80], [HD79], Delves3 [HDP79]), Delves4 who call it global element method. Its advantage is
the symmetry of the discrete problem. On the other hand, a significant disadvantage
is that the billinear form Bh s (·, ·) is indefinite. This causes troubles
when dealing with time-dependent problem since some eigenvalues can have
negative real parts and then the resulting formulations becames unconditionally
unstable. Therefore we prove in Lemma 2.18 lem:A11 the continuity of the bilinear
form Bh s , but further we shall not be concerned with this method any more.
The scheme ( 2.61) A2.22b was introduced by Baumann and Oden in [BBO99],
bbo
obb
[OBB98]. Therefore, we shall call it the Baumann-Oden method. It is straightforward
to show that the corresponding billinear form Bh n (·, ·) is positive
semidefinite. An interesting property of this method is that it is unstable
for piecewise linear approximations, i.e. for p = 1.