Unformatted text preview: tered di erencing of the two-dimensional Laplacian. 9.4 Krylov Subspace Methods
The conjugate gradient method is applicable to symmetric and positive de nite linear
systems. While many elliptic problems satisfy these conditions, there are important 9.4. Krylov Subspace Methods 55 problems that do not. Convection-di usion equations, studied in Section 4.4 are a good
example. The use of upwind di erencing of the convective ( rst-derivative) terms will
produce an unsymmetric algebraic system. Of course, if our goal is to solve (9.1.1), we
could modify the problem to AT Ax = AT b: (9.4.1) This system is symmetric and positive de nite hence, we can solve it using the steepest
descent or conjugate gradient method. However, the condition number (9.3.4c) of AT A
is the square of that of A, and this may lead to ill conditioning. Thus, we can expect
slow convergence when applying this procedure.
An alternate possibility is to seek an approximate solution by projection methods.
With such an approach, our goal is to nd approximations
x = x(0) + v (9.4.2a) where x(0) is an initial guess and v is an element of an M -dimensional (M N ) subspace
V of <N . The subspace V is called the trial, candidate, or search space. In order to
determine x we must specify M conditions or constraints for v to satisfy. Using the
Petrov-Galerkin method, we introduce a second M -dimensional subspace W of <N , and
determine x such that
(~ w) = (b ; Ax w) = 0
r 8w 2 W : (9.4.2b) Thus, the residual r is orthogonal to all (M linearly independent) vectors w 2 W . The
subspace W is called the test or constraint space. Orthogonality can involve the usual L2
(v w) := wT v (9.4.2c) however, we'll consider other inner products (e.g., strain energy (9.3.12a)) as well.
Using (9.4.2a) in (9.4.2b)
(b ; A(x(0) + v) w) = (r(0) ; Av w) = 0 8w 2 W : (9.4.3) 56 Solution Techniques for Elliptic Problems
Av r = r (0) Av
- r(0) W Figure 9.4.1: Orthogonal projection in three dimensions.
The orthogonality condition is now imposed on the \new residual" r(0) ; Av and all
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