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The discrete least squares method for 2m-th order elliptic boundary-value problems Sammon, Peter Henry

Abstract

Many positive results are known for the Least Squares method of numerically computing an approximant solution for a 2m-th order elliptic boundary value problem on a real interval. However due to integrals that appear in the linear system that must be solved to find the approximant solution, the Least Squares method is computationally unattractive. Thus one is led to consider other, more practical, methods. In this thesis, such a method is examined. It is first shown that minimizing a quantity involving discrete quadrature sums to find an approximation to the solution leads to a linear system in which quadrature sums, not integrals, appear. The approximation generated by this method, known as the Discrete Least Squares method, is thus easily computed. Error estimates are then proven for the Discrete method. A certain polynomial interpolate is first constructed and then using Sobolev space theory and known estimates from the related Least Squares method, the desired estimates are obtained. These estimates are like the usual Least Squares estimates although more continuity of the coefficients of the problem is required in the final stages. Comparisons are then drawn between the Discrete method and the Collocation method. It is noted that both methods offer easy computability and shown why the Discrete method can generate a smoother approximation in most situations. Results of numerical computations are then provided which support the theory and a discussion of what problems the theory applies to is also given.

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