SUMMARYThis paper presents a new method of formulating the finite element relationships based on the least squares criterion. To overcome the high degree of inter-element continuity, a method of reducing the original governing differential equation to a set of equivalent system of first-order differential equations is proposed. The validity of the method is demonstrated by means of several numerical examples. In particular, application of the method to problems with unknown variational functionals is considered.

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.

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