Multiwavelet approximation methods for pseudodifferential equations on curves. Stability and convergence analysis
- Prößdorf, Siegfried
- Schult, Jörg
2010 Mathematics Subject Classification
- 65J10 65N30 65N35 65R20 47G30 45P05 41A25 41A30 41A15
- Periodic pseudodifferential equations, multiwavelets, splines with multiple knots, generalized Galerkin-Petrov schemes, boundary element methods, error analysis, stability, Strang-Fix condition
We develop a stability and convergence analysis of Galerkin-Petrov schemes based on a general setting of multiresolution generated by several refinable functions for the numerical solution of pseudodifferential equations on smooth closed curves. Particular realizations of such a multiresolution analysis are trial spaces generated by biorthogonal wavelets or by splines with multiple knots. The main result presents necessary and sufficient conditions for the stability of the numerical method in terms of the principal symbol of the pseudodifferential operator and the Fourier transforms of the generating multiscaling functions as well as of the test functionals. Moreover, optimal convergence rates for the approximate solutions in a range of Sobolev spaces are established.
- Adv. Comput. Math., 9 (1998), pp. 145-171