Abstract
This chapter focuses on spline approximation and difference schemes for the heat equation. The chapter discusses some preliminaries concerning certain trigonometric polynomials related to the B-splines, and reviews some properties of the spline interpolation operator. The finite difference operator Fk is analyzed by studying its symbol. It is proved that it is parabolic and accurate of order, 2μ - 2. Certain mixed initial-boundary value problems can be rephrased as pure initial-value problems by periodicity considerations. The chapter presents an example of such a problem and shows how a specific discrete time Galerkin scheme can be applied to initial data with and without smoothing. Some well-known facts about Toeplitz operators on sequences are reviewed. The continuous time Galerkin method, and the application of the finite difference theory is discussed. The chapter also presents a class of discretizations in time and analyzes the corresponding difference schemes. Various lemmas are also reviewed.
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