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 $ L $ be an operator with domain of definition $ D ( L) $ in a Banach space $ X $ and with range $ R ( L) $ in a Banach space $ Y $. To solve the equation

$$ \tag{1 } Lx = y $$

by a projection method, one chooses two sequences $ \{ X _ {n} \} $ and $ \{ Y _ {n} \} $ of subspaces

$$ X _ {n} \subset D ( L) \subset X ,\ Y _ {n} \subset Y ,\ \ n = 1 , 2 \dots $$

as well as projectors $ P _ {n} $ projecting $ Y $ onto $ Y _ {n} $. Equation (1) is replaced by the approximate equation

$$ \tag{2 } P _ {n} L x _ {n} = P _ {n} y ,\ \ x _ {n} \in X _ {n} . $$

In the case $ X = Y $, $ X _ {n} = Y _ {n} $, $ n = 1 , 2 \dots $ 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 $ X _ {n} $ and $ Y _ {n} $). Suppose that $ L $ is linear and takes $ D ( L) $ onto $ R ( L) $ bijectively, with $ D ( L) $ and $ R ( L) $ dense in $ X $ and $ Y $, respectively. Suppose that the subspaces $ X _ {n} $ and $ Y _ {n} $ are finite-dimensional, $ \mathop{\rm dim} X _ {n} = \mathop{\rm dim} Y _ {n} $, $ n = 1 , 2 \dots $ and that the projectors $ P _ {n} $ are uniformly bounded in $ n $, that is, $ \| P _ {n} \| \leq \textrm{ const } $, $ n = 1 , 2 , . . . $. Then the following condition a) is equivalent to the conditions b) and c) combined.

a) From some $ n = n _ {0} $ onwards there exists a unique solution $ x _ {n} $ of (2), and $ \| L x _ {n} - y \| \rightarrow 0 $ for any $ y \in Y $;

b) The sequence of subspaces $ L X _ {n} $ is dense in the limit in $ Y $, that is, the distance $ d ( y , L X _ {n} ) \rightarrow 0 $ as $ n \rightarrow \infty $ for every $ y \in Y $;

c) $ \tau \equiv \lim\limits _ {n \rightarrow \infty } \tau _ {n} > 0 $, where $ \tau = \inf \{ {\| P _ {n} y _ {n} \| } : {y _ {n} \in L X _ {n} , \| y _ {n} \| = 1 } \} $.

The rate of convergence under the conditions b) and c) is characterized by the inequality

$$ \tag{3 } d ( y , L X _ {n} ) \leq \| L x _ {n} - y \| < \left ( 1 + \frac{c}{\tau _ {n} } \right ) d ( y , L X _ {n} ) . $$

If $ X $ and $ Y $ are Hilbert spaces and $ P _ {n} $ and $ Q _ {n} $ are orthoprojectors projecting $ Y $ onto $ Y _ {n} $ and $ L X _ {n} $, respectively, condition c) is equivalent to the condition

c') $ \theta \equiv {\overline{\lim\limits}\; } _ {n \rightarrow \infty } \theta _ {n} < 1 $, where $ \theta _ {n} = \| P _ {n} - Q _ {n} \| $ is the gap (angle) between the subspaces $ Y _ {n} $ and $ L X _ {n} $; instead of (3) one obtains the estimate

$$ \| y - Q _ {n} y \| \leq \| L x _ {n} - y \| \leq \ \frac{1}{\sqrt {1 - \theta _ {n} ^ {2} }} \| y - Q _ {n} y \| . $$

In case $ Y _ {n} = L X _ {n} $( the least-squares method) one has $ \theta _ {n} = 0 $, $ n = 1 , 2 \dots $ and the convergence criterion is the condition b).

The theorem yields a condition for convergence of the discrepancy $ \| L x _ {n} - y \| $. If $ L ^ {-1} $ is bounded and $ y \in R ( L) $, then convergence of the discrepancy implies convergence of the approximations $ x _ {n} $ themselves to the solution $ x = L ^ {-1} y $ 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 $ l $ is a bounded linear form and $ a $ is a bounded bilinear form on a real Hilbert space $ H $( or sesquilinear in the case of a complex $ H $). It is assumed that $ a $ is representable as $ a = \widehat{a} + b $, such that

$$ \widehat{a} ( u , u ) \geq \gamma \| u \| ^ {2} \ \ \forall u \in H ,\ \dot \gamma = \textrm{ const } > 0 , $$

while the bilinear form $ b $ is completely continuous, i.e. the weak convergences $ u _ {n} \rightarrow u $, $ v _ {n} \rightarrow v $ in $ H $ imply the convergence $ b ( u _ {n} , v _ {n} ) \rightarrow b ( u , v ) $( the forms $ a , \widehat{a} , b $ are not necessarily symmetric). Suppose that the following problem is posed: Find a $ u \in H $ such that

$$ \tag{4 } a ( u , v ) = l ( v) \ \ \forall v \in H . $$

The Galerkin method for solving (4) consists in the following. One chooses (closed) subspaces $ H _ {n} \subset H $, $ n = 1 , 2 \dots $ and finds a $ u _ {n} \in H _ {n} $ such that

$$ \tag{5 } a ( u _ {n} , v _ {n} ) = l ( v _ {n} ) \ \ \forall v _ {n} \in H _ {n} . $$

The following theorem holds: Suppose that $ \{ H _ {n} \} $ is dense in the limit in $ H $, that the conditions imposed above on $ a $ are satisfied and that problem (4) has a unique solution $ u \in H $( an equivalent condition is: the homogeneous problem of finding $ u $ from the condition $ a ( u , v ) = 0 $ for every $ v \in H $ has only the trivial solution $ u = 0 $); then problem (5) for sufficiently large $ n $ has a unique solution $ u _ {n} \in H _ {n} $, and $ \| u _ {n} - u \| \rightarrow 0 $ with the estimate

$$ \| u - O _ {n} u \| \leq \| e _ {n} - u \| \leq \ c \| u - O _ {n} u \| , $$

where $ O _ {n} $ is the orthoprojector projecting $ H $ onto $ H _ {n} $ and $ c = \textrm{ const } $.

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 $ H $.

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)

References

[a1]	C.A.J. Fletcher, "Computational Galerkin methods" , Springer (1984)