Projection Methods for Variational Equations
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In this chapter, we reformulate the examples presented in Section 1.1 as variational equations, and then derive methods for approximating their solutions. In Section 2.1, we establish the relation between a linear operator equation and the associated variational formulation whenever the underlying linear space is a prehilbert space and then derive variational formulations in detail for each of the sample problems in Section 1.1. In order to deduce, however, existence and uniqueness of solutions to our examples via the fundamental Lax-Milgram Lemma, we must require our underlying spaces to be Hilbert spaces. Consequently, in Section 2.2, we complete to a Hilbert space each of the prehilbert spaces associated with the respective problem and discuss the concept of a generalized (or weak) solution. We then approximate the solutions of our variational problems in Sections 2.3 and 2.4 by considering and solving each problem in a finite-dimensional subspace of the respective Hilbert spaces obtained in Section 2.2. This procedure can be viewed as a projection method. Among the many special types of projection methods in this chapter, Ritz-Galerkin methods are used for approximating solutions to the linear examples. In Section 2.5, we shall see that projection methods can just as well be used to approximate solutions to nonlinear problems. These methods result in a nonlinear system of equations which must be solved iteratively, e.g., by Newton methods. We conclude this chapter by presenting a projection method for the nonlinear boundary-value problem introduced in Section 2.2 and show how approximations can be constructed by a procedure analogous to the Ritz-Galerkin method.
KeywordsHilbert Space Bilinear Form Projection Method Variational Equation Positive Semidefinite
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