Projection methods
Methods for finding an approximate solution of an operator equation in a prescribed subspace, based on projecting the equation onto some (generally speaking, different) subspace. Projection methods constitute the basis for various computational schemes for solving boundary value problems, including the finite element and collocation methods (cf. Galerkin method; Collocation method).
Let be an operator with domain of definition
in a Banach space
and with range
in a Banach space
. To solve the equation
![]() | (1) |
by a projection method, one chooses two sequences and
of subspaces
![]() |
as well as projectors projecting
onto
. Equation (1) is replaced by the approximate equation
![]() | (2) |
In the case ,
,
the projection method (2) is usually called the Galerkin method (sometimes the latter method is interpreted in a wider sense, see Galerkin method).
A convergence theorem holds for projection methods for linear equations (in the case of finite-dimensional subspaces and
). Suppose that
is linear and takes
onto
bijectively, with
and
dense in
and
, respectively. Suppose that the subspaces
and
are finite-dimensional,
,
and that the projectors
are uniformly bounded in
, that is,
,
. Then the following condition a) is equivalent to the conditions b) and c) combined.
a) From some onwards there exists a unique solution
of (2), and
for any
;
b) The sequence of subspaces is dense in the limit in
, that is, the distance
as
for every
;
c) , where
.
The rate of convergence under the conditions b) and c) is characterized by the inequality
![]() | (3) |
If and
are Hilbert spaces and
and
are orthoprojectors projecting
onto
and
, respectively, condition c) is equivalent to the condition
c') , where
is the gap (angle) between the subspaces
and
; instead of (3) one obtains the estimate
![]() |
In case (the least-squares method) one has
,
and the convergence criterion is the condition b).
The theorem yields a condition for convergence of the discrepancy . If
is bounded and
, then convergence of the discrepancy implies convergence of the approximations
themselves to the solution
of equation (1). From the theorem, a convenient criterion for the convergence of the Galerkin method can be extracted; for the Galerkin–Petrov method an additional condition of the type c') should be imposed.
Suppose that is a bounded linear form and
is a bounded bilinear form on a real Hilbert space
(or sesquilinear in the case of a complex
). It is assumed that
is representable as
, such that
![]() |
while the bilinear form is completely continuous, i.e. the weak convergences
,
in
imply the convergence
(the forms
are not necessarily symmetric). Suppose that the following problem is posed: Find a
such that
![]() | (4) |
The Galerkin method for solving (4) consists in the following. One chooses (closed) subspaces ,
and finds a
such that
![]() | (5) |
The following theorem holds: Suppose that is dense in the limit in
, that the conditions imposed above on
are satisfied and that problem (4) has a unique solution
(an equivalent condition is: the homogeneous problem of finding
from the condition
for every
has only the trivial solution
); then problem (5) for sufficiently large
has a unique solution
, and
with the estimate
![]() |
where is the orthoprojector projecting
onto
and
.
In applications to boundary value problems for equations of elliptic type, as a rule, the energy space of the principal part of the corresponding differential operator is chosen as .
References
[1] | M.A. Krasnosel'skii, G.M. Vainikko, P.P. Zabreiko, et al., "Approximate solution of operator equations" , Wolters-Noordhoff (1972) (Translated from Russian) |
Comments
References
[a1] | C.A.J. Fletcher, "Computational Galerkin methods" , Springer (1984) |
Projection methods. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Projection_methods&oldid=48312