Convergence Analysis Of Methods For Solving
General Equations
Author(s)
R. Schaback
Abstract
The unsymmetric collocation method by E. Kansa has been very successfully used
in many applications, though there theoretically exist rare situations in which it
will fail. This contribution modifies the method somewhat and then proves convergence
and error bounds for the modified technique. The results will be presented
within a general framework for methods that solve operator equations by minimizing
residuals. Thus the modifications may also help to put other methods on a solid
foundation.
1 Linear operator equations
We consider a linear problem
D(u) = w (1)
which has a solution u that has to be recovered from data w = D(u) under a linear
mapping D. This setting covers many differential or integral equation problems in
weak or strong form. For convenience, we restrict ourselves here to examples of
problems in strong form.
In case of a Poisson boundary value problem
−∆u = f in Ω
u = ϕ on ∂Ω
(2)
the data w = D(u) for a strong problem formulation will be
w = (f,ϕ) = D(u) := (−∆, Id∂Ω)(u).
Keywords
Related Book
Other papers in this volume
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