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In the present chapter, we derive a spectral Galerkin semidiscretization of Problems 2.21 and 2.22 in which we choose the eigenfunctions of the operator \(\mathcal {H}\) as trial functions. This is preceded by an investigation of the eigenfunctions in view of their approximation properties. In particular, an eigenvalue estimate derived from Weyl’s law allows an estimation of the truncation error. Further, we discuss the approximation error of the spectral Galerkin semidiscretization as well as the solution of the evolving system of ODEs by exponential integrators. Finally, we examine two particular aspects of implementation and give an outlook on the application to IBVP governed by other classes of PDEs, such as a fractional diffusion equation.

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Spectral Solution Method

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  • 发表日期：

    2025年

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  • Anna Weller

摘要

In the present chapter, we derive a spectral Galerkin semidiscretization of Problems 2.21 and 2.22 in which we choose the eigenfunctions of the operator \(\mathcal {H}\) as trial functions. This is preceded by an investigation of the eigenfunctions in view of their approximation properties. In particular, an eigenvalue estimate derived from Weyl’s law allows an estimation of the truncation error. Further, we discuss the approximation error of the spectral Galerkin semidiscretization as well as the solution of the evolving system of ODEs by exponential integrators. Finally, we examine two particular aspects of implementation and give an outlook on the application to IBVP governed by other classes of PDEs, such as a fractional diffusion equation.