Convergence Analysis of Special Methods
In this chapter, we analyze the behavior of the truncation errors for various methods approximating solutions of linear and nonlinear initial value problems. Based on this analysis, and on results on inverse stability from Chapter 12, convergence results for these methods are then obtained. It is appropriate at this point to emphasize that the investigation of the truncation errors and the resulting convergence analysis must be carried out with respect to those norms for which inverse stability is assured for the method being considered, and — for nonlinear problems — for which the uniform differentiability conditions in Section 11.1 are satisfied. For finite-difference methods, the truncation errors will be examined by using Taylor series expansions; then convergence along with error estimates will be shown with respect to both the discrete supremum norms and the discrete L2-norms. For Galerkin methods, we aim to produce quasi-optimal error estimates with respect to the approximations in the spatial variable. Attaining such estimates requires extensive investigations for rather simple problems so that we shall necessarily restrict the scope of our analysis of Galerkin methods to an example for the heat and the wave equations, respectively.
KeywordsGalerkin Method Special Method Convergence Analysis Truncation Error Supremum Norm
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References (cf. also References in Chapters 4, 11 and 12)
- Ansorge & Hass (1970), Aubin (1979), Bramble & Sammon (1980)*, Ciarlet (1978), Dahlquist (1956)*, Davis (1963), Dupont (1973)*, Fairweather (1978), Mitchell & Griffiths (1980), Törnig (1963)*, Wahlbin (1978)*.Google Scholar