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from Part V - Boundary and Initial Boundary Value Problems
Published online by Cambridge University Press: 29 September 2022
Collocation methods for elliptic problems are discussed here. We begin by providing their definition. For their analysis we first introduce a weighted weak formulation of the problem, and show that it is well posed. Then, we introduce and analyze a Galerkin approximation for this problem, where the subspace consists of polynomials that vanish sufficiently fast at the boundary. Next, a scheme with quadrature is proposed, and its analysis is provided using the theory of variational crimes and Strang lemmas. For its implementation and analysis the discrete cosine and Chebyshev transforms are introduced and analyzed. The phenomenon of aliasing is briefly discussed. Finally, we connect the weighted Galerkin approximation with quadrature to collocation methods, thus providing an analysis of collocation schemes.
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