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Mathematics of Computation

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A mollifier useful for approximations in Sobolev spaces and some applications to approximating solutions of differential equations

Author: Stephen Hilbert
Journal: Math. Comp. 27 (1973), 81-89
MSC: Primary 65D10; Secondary 65N99
MathSciNet review: 0331715
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Abstract: For a given uniform grid of $ {E^N}$ (N-dimensional Euclidean space) with mesh h, a class of smoothing functions (mollifiers) is constructed. If a function is an element of the Sobolev space $ H_2^m$, then the error made by replacing the given function by a smoother $ ({C^\infty })$ function (which is the given function convolved with one of the mollifiers) is bounded by a constant times $ {h^m}$.

This result is used to construct approximations for functions using Hermite or spline interpolation, even though the function to be approximated need not satisfy the continuity conditions necessary for the existence of a Hermite or spline interpolate. These techniques are used to find approximations to the generalized solution of a second order elliptic Neumann problem.

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Keywords: Finite element method, Galerkin method, Rayleigh-Ritz method, Hermite interpolation, spline, mollifier, approximation, Sobolev space, boundary value problem
Article copyright: © Copyright 1973 American Mathematical Society

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