Mathematical Foundations of Least-Squares Finite Element Methods
In Section 2.2, we introduced many of the ideas that form the core of modern leastsquares finite element methods (LSFEMs). In this chapter, we develop a mathematical theory that makes precise the key ideas and provides a rigorous framework for the application of least-squares principles. At the center of our framework is an abstract least-squares theory for solving operator equations in Hilbert spaces. The specialization of this framework to partial differential equation (PDE) problems provides a template for LSFEMs that is used throughout the book.
The remainder of the chapter is devoted to the formulation and analysis of the discrete least-squares principle (DLSPs) that define least-squares finite element approximations of the solution of the PDE problem. We present the theory in two stages. The first stage (see Section 3.3) examines what can be expected from a discrete residual minimization principle if no connection to a CLSP class is assumed. This stage not only helps to explain the remarkable robustness of LSFEMs, but also reveals the limitations of this very general setting. In the second stage, the analysis is extended to include DLSPs obtained from a CLSP class associated with the PDE problem. In Section 3.4, we examine the transformation of a CLSP into a DLSP and show that this process consists of choosing approximate norm-generating and differential operators.2 In Section 3.5, the three basic types of DLSPs, e.g., compliant, norm-equivalent, and quasi-norm-equivalent, are shown to result from specific approximation choices. Using the link between CLSPs and DLSPs, we develop there an approximation theory for least-square finite element approximations of solutions of PDE problems.
KeywordsTruncation Error Residual Energy Energy Norm Mathematical Foundation Auxiliary Problem
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