Finite elements formulated by the weighted discrete least squares method

Abstract

SUMMARYBy use of the least squares error criterion, an alternate finite element formulation is presented. The method is based on the discrete or element-wise minimization of square and weighted differential equation residuals which are expressed in terms of element nodal quantities. In order to overcome the stringent inter-element continuity requirement, a major stumbling block, on the element trial functions two practical schemes are proposed. One is the reduction of the original governing differential equation to a system of equivalent firstorder differential equations ; the other is a method of smoothing discontinuous trial functions. The latter essentially relaxes the continuity requirement and yields efficient non-conforming finite elements. This paper also demonstrates the use of constant weights which significantly improves the rates of convergence. Several numerical examples illustrate the proposed method. From these examples, it may be concluded that the use of constant weights and the relaxation of the inter-element continuity requirement are two indispensable features of the weighted discrete least squares method.

A note on the least squares solution of the groundwater flow equation

“…Very refined meshes are required to match solution accuracies of the Galerkin and mixed least squares finite element methods. The least squares finite element method may perform better if the interelement continuity requirements were relaxed by, for example, applying penalty methods [6] at the element interfaces.…”

“…Unless the mixed formulation [6,7] or special smoothing procedures are used,6 the least squares finite element method requires the use of a C' continuous trial space. Hydraulic head is approximated with a trial function it = C U~Q~, (2) where CYJ are time varying coefficients and, in the examples that follow, QJ are C' continuous cubic Hermite basis functions.…”

“…Unless the mixed formulation [6,7] or special smoothing procedures are used,6 the least squares finite element method requires the use of a C' continuous trial space. Hydraulic head is approximated with a trial function…”

Stress separation of thermoelastically measured isopachics

“…However, others have used nodal points or evenly distributed locations throughout an element with excellent results. [23][24][25] Assuming one has isopachic data S, the dependent variables in eq (i) are cry and Xxy. An approximate solution to the differential equations begins by selecting admissible functions for these variables.…”

On the solution of the advection equation and advective dominated reactor models by weighted residual methods

“…The least-squares method began to appear in the literature in the late 1960's [8][9][10][11][12]. In the last decades, the least-squares method has been applied to solve a variety of different problems, e.g., the Stokes equations [13][14][15][16], Euler equations [17], Navier-Stokes equations [18][19][20][21], electro-magnetism [22], viscoelastic flows [23], Burgers equation [24] and sound propagation [25,26].…”