Semidiscrete least-squares methods for a parabolic boundary value problem

In this paper some approximate methods for solving the initial-boundary value problem for the heat equation in a cylinder under homogeneous boundary condi- tions are analyzed. The methods consist in discretizing with respect to time and solving approximately the resulting elliptic problem for fixed time by least squares methods. The approximate solutions will belong to a finite-dimensional subspace of functions in space which will not be required to satisfy the homogeneous boundary conditions. 1. Introduction. The purpose of this paper is to analyze some approximate methods for solving the initial-boundary value problem for the heat equation in a cylinder under homogeneous boundary conditions. The methods consist in discretizing with respect to time and solving approximately the resulting elliptic problem for fixed time by least squares methods. The approximate solutions will belong to a finite- dimensional subspace of functions in space which will not be required to satisfy the homogeneous boundary conditions. Let Ql be a bounded domain in Euclidean N-space RN with smooth boundary au. We shall consider the approximate solution of the following mixed initial-boundary value problem for u = u(x, t), namely,

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