The Approximate Solution of Parabolic Initial Boundary Value Problems by Weighted Least-Squares Methods - King, J. Thomas (SIAM journal on numerical analysis, 1972)
This paper is primarily concerned with the problem of finding approximants for the solution of initial boundary value problems of parabolic type and obtaining a priori error estimates. Perhaps the most widely used method for the treatment of such problems is that of finite differences. Some difficulties arising in this method are stability of the scheme and the treatment of the boundary. Another approach to this problem is the so-called semidiscrete variational method. As in the finite difference approach, a difficulty in this method is the treatment of boundary values. The methods considered in this paper will be of the least-squares type. Roughly speaking, the method consists in finding that element of a suitable finite-dimensional subspace which best approximates the data of the initial-boundary value problem in a least-squares sense. A principal virtue of this approach is that the elements of the finite-dimensional subspace are not required to satisfy any boundary or initial conditions. A particular choice of the finite-dimensional subspace could be a certain class of mixed homogeneous splines. Results are obtained which show that, for a particular least-squares scheme, the error made in approximating the solution, u, of the initial boundary value problem by the least-squares approximant is the same (modulo a constant) as the error made in approximating u by any element of the finite-dimensional subspace. The proofs of the error estimates are based on the theory of interpolation spaces and the theory of approximation in Sobolev spaces.