Civil Engineering Reference
In-Depth Information
0.
Its construction heavily depends on the following assertion. The functions in
(
P
1
)
2
which have the form
For convenience, we consider the Raviart-Thomas element only for
k
=
a
b
+
c
x
p
=
y
are characterized by the fact that
n
·
p
is constant on each line
αx
+
βy
=
const
whenever
n
is orthogonal to the line. Therefore, given a triangle
T
, the normal
component is constant and can be prescribed on each edge of
T
(see Fig. 35).
Formally, the Raviart-Thomas element is the triple
(T , (
P
0
)
2
+
x
·
P
0
,n
i
p(z
i
), i
=
1
,
2
,
3 with
z
i
being the midpoint of edge
i).
The solvability of the interpolation problem is easily verified. Given a vertex
a
i
of
T
, we can find a vector
r
i
∈ R
2
such that its projections onto the normals
of the adjacent edges have the prescribed values. Now determine
p
∈
(
P
1
)
2
such
that
p(a
i
)
=
r
i
,i
=
1
,
2
,
3
.
It is immediate from
p
∈
(
P
1
)
2
that the normal components are linear on each
edge of the triangle. They are even constant, since by construction they attain
the same value at both vertices of the edge. Thus the function constructed indeed
belongs to the specified subset of
(
P
1
)
2
.
A proof of the inf-sup condition will be given below.
The Raviart-Thomas element and the similar BDM elements due to Brezzi,
Douglas, and Marini [1985] are frequently used for the discretization of prob-
lems in
H(
div
,)
. Analogous elements for 3-dimensional problems have been
described by Brezzi, Douglas, Duran, and Fortin [1987].
The finite element solution of the Raviart-Thomas element is related to the
nonconforming
P
1
element; see Marini [1985].
Interpolation by Raviart-Thomas elements
Due to Theorem 4.5 the error of the finite element solution for the discretization
with the Raviart-Thomas element can be expressed in terms of approximation
by the finite element functions. As usual the latter is estimated via interpolation.
To this end an interpolation operator is defined which is based on the degrees of
freedom specified in the definition of the element.
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