Keele Research Repository

ii.is thesis is concerned with dual extremum principles and their applications. Dual extremum principles provide bounds to a functional associated with a given equation, and in some cases this functional is a measure oi a physical quantity. These bounds can sometimes also provide information about the solution of the equation.
Chapter I is a survey of dual extremum principles from the classical maximum and minimum principles. Three important applications are summarised in this section.
Chapter II, on general principles, forms the basis of the thesis and includes definitions and theorems necessary for the following chapters. Sections on differential and integral operators, and convergence of iterative schemes, ends the chapter.
Chapter III develops optimising iterative schemes for dual extremum principles for linear problems. Convergence of the iterative schemes is considered and two examples complete the chapter.
In Chapter IV, we look at the decomposition of saddle functionals into two saddle functionals with a view to obtaining improved bounds. The method is applied to a particular functional, and conditions are found which ensure that the decomposition bounds are sharper than the classical bounds. The combination of iterative methods and decomposition dual extremum principles is considered, and conditions for convergence are found. The chapter ends with four examples.
Chapter V deals with comparison functionals, a method which involves iinding simpler saddle functionals to approximate the saddle functional for which bounds are required. The theory is applied to the same particular functional as that used in chapter IV, and conditions are developed which ensure that the comparison bounds are sharper than the classical bounds. The combination of comparison bounds and iterative methods is incluaec., and conditions for convergence are found. Two examples end the chapter.
Chapter VI is a short chapter which looks at dual extremum principles for functionals which are convex/concave but not saddle over the whole of their domain. Three applications of the theory are included.