On least squares approximations to indefinite problems of the mixed type

Abstract

SUMMARYA least squares method is presented for computing approximate solutions of ipdefinite partial differential equations of the mixed type such as those that arise in connection with transonic flutter analysis. The method retains the advantages of finite difference schemes namely simplicity and sparsity of the resulting matrix system. However, it offers some great advantages over finite difference schemes. First, the method is insensitive to the value of the forcing frequency, i.e., the resulting matrix system is always symmetric and positive definite. As a result, iterative methods may be successfully employed to solve the matrix system, thus taking full advantage of the sparsity. Furthermore, the method is insensitive to the type of the partial differential equation, i.e., the computational algorithm is the same in elliptic and hyperbolic regions. In this work the method is formulated and numerical results for model problems are presented. Some theoretical aspects of least squares approximations are also discussed.

The Least-Squares Finite Element Method

“…This method, which represents the beginning of LSFEM for first-order systems, was proposed in the pioneering papers of Lynn and Arya (1973) and Zienkiewicz et al (1974) and later developed and investigated by Fix and Gunzburger (1978), Fix et al (1979), and Cox and Fix (1984). The conventional LSFEM has a significant advantage over the mixed Galerkin method 112 6.…”

Finite Element Methods of Least-Squares Type

“…As a result, least-squares finite element methods in these settings are among the best understood, studied, and tested from both the theoretical and computational viewpoints. Our discussion will also include least-squares methods for convection-diffusion and other second-order elliptic problems (see [6], [24], [26], [33], [35], [38], [42]- [45], [48], [52], [62], [71], [72], [75], [86], [95], [106], [107], and [104]), linear elasticity (see [34], [36], and [37]), inviscid, compressible flows (see [61], [64], [67], [99], and [118]), and electromagnetics (see [46], [58], [60], [97], and [116]. )…”

A Computational Study of Stabilized, Low-order C 0 Finite Element Approximations of Darcy Equations

“…The stabilized method (29) can be viewed as a weighted average of the unstable mixed method (13) and (14) and the stable Galerkin method (27). Alternatively, we can view (29) as obtained by adding the weak residual…”

“…In this paper we focus on four different possibilities to accomplish this. The first one is to use the classical least-squares approach for the first-order system (1), (2), and (3), formulated in [14,15,18]. The second is to stabilize the mixed problem by using the residual of (2).…”