Solvability of partial differential equations by meshless kernel methods
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Solvability of partial differential equations by meshless kernel methods Y. C. Hon · Robert Schaback
Received: 6 March 2006 / Accepted: 16 April 2006 / Published online: 6 December 2006 © Springer Science+Business Media B.V. 2006
Abstract This paper first provides a common framework for partial differential equation problems in both strong and weak form by rewriting them as generalized interpolation problems. Then it is proven that any well-posed linear problem in strong or weak form can be solved by certain meshless kernel methods to any prescribed accuracy. Keywords solvability · kernel · meshless · partial differential equations Mathematics Subject Classifications (2000) 35A25 · 65N35
1. Linear problems The fairly general statement made in the abstract needs some specification. We assume a problem to be posed that is solvable by a function u in some Hilbert space U with inner product (·, ·)U . Note that this is satisfied for all problems that can be formulated in Sobolev spaces, for instance, but we also allow problems with strong solutions in Hilbert subspaces of differentiable or Hölder continuous functions. The elements of U are viewed as multivariate functions, and the elements λ ∈ U ∗ are
The work described in this paper was partially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 101205). Robert Schaback’s research in Hong Kong was sponsored by DFG and City University of Hong Kong. Y. C. Hon (B) Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong e-mail: [email protected] R. Schaback Institut für Numerische und Angewandte Mathematik, Lotzestraße 16–18, D37083 Göttingen, Germany
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Adv Comput Math (2008) 28:283–299
continuous linear functionals that we use to describe data λ(u) of u, e.g. evaluations u 7→ δx (u) := u(x) or u 7 → (δx ◦ 1)(u) = (1u)(x). The problems should be formulated by requiring that a (usually uncountable) set 3 of functionals, when applied to the solution u, attains certain prescribed values. This means that u solves a (usually uncountable) set of equations λ(u) = ϕ(λ) for all λ ∈ 3
(1)
where ϕ : 3 → IR is a given function. We do not care about assumptions on ϕ, but we assume that the functionals λ ∈ 3 are continuous on U, i.e. they must be in the dual U ∗ of U. We call them test functionals, because they define the test criteria for a function u to be a solution of our problem. It is shown in the next section that plenty of strongly or weakly formulated linear problems of Applied Analysis have this form, because the test functionals λ can, for instance, describe point evaluations of u, its derivatives, or some differential or integral operator applied to u. Definition 1. A problem (1) is admissible, if it is posed with 3 ⊆ U ∗ , ϕ : 3 → IR and solvable by some function u ∈ U. An admissible problem will have a unique solution in U, if we know that the closed linear subspace of homogeneous solutions consists of the zero function only, but we need not assu
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