Projection methods

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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


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


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


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


The Galerkin method for solving (4) consists in the following. One chooses (closed) subspaces , and finds a such that


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 .


[1] M.A. Krasnosel'skii, G.M. Vainikko, P.P. Zabreiko, et al., "Approximate solution of operator equations" , Wolters-Noordhoff (1972) (Translated from Russian)



[a1] C.A.J. Fletcher, "Computational Galerkin methods" , Springer (1984)
How to Cite This Entry:
Projection methods. G.M. Vainikko (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098