Spline reproducing kernels on and error bounds for piecewise smooth LBV problems
Abstract Reproducing kernel method for approximating solutions of linear boundary value problems is valid in Hilbert spaces composed of continuous functions, but its convergence is not satisfactory without additional smoothness assumptions. We prove 2nd order uniform convergence for regular problems with coefficient piecewise of Sobolev class H2. If the coefficients are globally of class H2, more refined phantom boundary NSC-RKHS method is derived, and the order of convergence rises to 3 or 4, according to whether the problem is piecewise of class H3 or H4. The algorithms can be successfully applied to various non-local linear boundary conditions, e.g. of simple integral form. The paper contains also a new explicit formula for general spline reproducing kernels in Hm[a, b], if the inner product 〈 f , g 〉 m , ξ = ∑ i m f ( i ) ( ξ ) g ( i ) ( ξ ) + ∫ f ( m ) g ( m ) depends on any fixed reference point ξ ∈ [a, b]. The piecewise NSC–RKHS methods are then applied to two example regular LBV problems in H3 and H5. Exactness of the resulting numerical solutions, the degree of convergence, and their dependency of the reference point ξ ∈ [a, b] are presented in attached figures.